Nonlinear Second Order Elliptic Equations Involving Measures

De Gruyter Series in Nonlinear Analysis and Applications 21 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Kitakyushu, Japan Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Boris N. Sadovsky, Voronezh, Russia Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Moshe Marcus Laurent Véron

Nonlinear Second Order Elliptic Equations Involving Measures

De Gruyter

Mathematics Subject Classification 2010: Primary: 35-02, 35J61, 35R06, 35J25, 35J91; Secondary: 28A33, 31A05, 46E35. Authors Prof. Dr. Moshe Marcus Technion – Israel Institute of Technology Dept. of Mathematics Technion City 32000 Haifa Israel [email protected] Prof. Dr. Laurent Véron Laboratoire de Mathematiques CNRS UMR 6083 Faculte des Sciences et Techniques Universite Francois Rabelais Parc de Grandmont 37200 Tours France [email protected]

ISBN 978-3-11-030515-9 e-ISBN 978-3-11-030531-9 Set-ISBN 978-3-11-030532-6 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P TP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

In the last 40 years semilinear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as: problems of physics and astrophysics, problems of differential geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions. An important family of such equations is that involving an absorption term, the model of which is u C g.x, u/ D 0 where ug.x, u/  0. Such equations are of particular interest because in them we have two competing effects: the diffusion expressed by the linear differential part and the absorption produced by the nonlinearity g. Furthermore, equations of this type with power nonlinearities play a crucial role in the study of superdiffusions. Naturally, the study of semilinear problems is based on linear theory and in particular on the theory of boundary value problems with L1 and, more generally, measure data. In addition to the classical theory of the Laplace equation, this study requires certain ideas of harmonic analysis such as the Herglotz theorem on boundary trace of positive harmonic functions and the resulting integral representation, Kato’s lemma and the boundary Harnack principle. These topics and their application to boundary value problems are treated in the first chapter. In the second chapter we turn to the main topic of this monograph: boundary value problems for the semilinear problem u C g.x, u/ D f uDh

in  on @

(1)

where f and h are L1 functions or more generally measures. Generally we assume that t 7! g., t / is a continuous mapping from R into L1.; /, where .x/ D d ist .x, @/, that g.x, / is non-decreasing for every x 2  and g.x, 0/ D 0. (L1 .; / denotes the weighted Lebesgue space with weight .) In addition we assume that lim g., t /=t D 1

t !1

(2)

uniformly with respect to x in compact subsets of . Two standard examples: g.x, t / D .x/ˇ jt jq1 t ,

g.x, t / D exp t  1.

(3)

vi

Preface

The problem (1) is understood in a weak sense; we require that u 2 L1./ and g ı u 2 L1.; /, that the equation holds in the distribution sense and that the data is attained in a weak sense, related to weak convergence of measures. In addition it is assumed that f 2 L1.; / or, more generally, f D  2 M.; /, i.e.,  is a Borel measure in  such that Z  d jj < 1. 

For the boundary data, it is assumed that h 2 L1 .@/ or, more generally, h D  2 M.@/, i.e.,  is a finite Borel measure on @. Problems with L1 data are discussed in Section 2.1. In this case the boundary value problem possesses a unique solution u 2 L1./ such that g ı u 2 L1.; / for every f 2 L1.; / and h 2 L1 .@/. An interesting feature of boundary value problems with measure data is that, in general, the problem is not solvable for every measure. If (1) has a solution for h D 0 and a measure f D  2 M.; /, we say that  is g-good in . The space of such measures is denoted by Mg .; /. Similarly, if (1) has a solution for f D 0 and a measure h D  2 M.@/, we say that  is g-good on @. The space of such measures is denoted by Mg .@/. If Mg .; / D M.; / we say that the nonlinearity g is subcritical in the interior. Similarly, if Mg .@/ D M.@/ we say that g is subcritical relative to the boundary. In Section 2.2 we present basic results on boundary value problems with measures. For instance, assuming that  and  are g-good, we show that (1) with f D , h D  has a unique weak solution u and derive estimates for kukL1./ and kg ı ukL1 .;/ in terms of the norms of  and  in their respective spaces. In particular we find that, if a solution exists it is unique. An important tool in our study is an extension of the method of sub- and supersolutions to the case of weak solutions and a general class of nonlinearities. This too is presented in Section 2.2 In Section 2.3 we present a sufficient condition for interior and boundary subcriticality. It is shown that this condition also implies stability with respect to weak convergence of data. Further, in Section 2.4, we discuss the structure of the space of good measures when the nonlinearity g is supercritical in the interior (resp. on the boundary), i.e., Mg .; /   M.; / (resp. Mg .@/   M.@/). Chapter 3 is devoted to a study of the boundary trace problem for positive solutions of the equation u C g.x, u/ D 0, (4) with g as in (1), and related boundary value problems. The basic model for our study is the boundary trace theory for positive harmonic functions due to Herglotz. By Herglotz’s theorem any positive harmonic function in a bounded Lipschitz domain admits a boundary trace expressed by a bounded measure and the harmonic function is uniquely determined by this trace via an integral representation.

Preface

vii

The notion of a boundary trace of a function u in  depends on the regularity propN then it has a boundary trace in C.@/, erties of the function. For instance, if u 2 C./ namely, ub@ . If u belongs to a Sobolev space W 1,p ./ for some p > 1 then it has a 1 boundary trace in Lp .@/ (and even in a more regular space, namely, W 1 p ,p .@/). The measure boundary trace of a positive harmonic function is defined as follows: let ¹n º be an increasing sequence of domains converging to ; under some restrictions on this sequence it can be shown that the sequence of measures ¹ub@n dSº converges N ( = the space of finite Borel measures in ) N to a measure  2 M.@/ weakly in M./ that is independent of ¹n º. This limiting measure is the measure boundary trace of u. If  is of class C 2 the harmonic function u can be recovered from its measure boundary trace via the Poisson integral. If the domain is merely Lipschitz, the Poisson kernel must be replaced by the Martin kernel. (For more details see Section 1.3.) As a first step in our study of the trace problem for positive solutions of (4) we consider moderate solutions. A positive solution of (4) is moderate if it is dominated by a harmonic function. The following result is a consequence of the Herglotz theorem. A positive solution u is moderate if and only if g ı u 2 L1 .; /. Every positive moderate solution possesses a boundary trace represented by a bounded measure. So far the trace problem for positive solutions of the nonlinear equation appears to be similar to the trace problem for positive harmonic functions. However, beyond this similarity, the nonlinear problem presents two essentially new aspects. The first is a fact already mentioned before: in general, there exist positive finite measures on @ that are not boundary traces of any solution of (4). The second: the equation may have positive solutions that do not have a boundary trace in M.@/. Both aspects are present in the basic examples (3). In the case of power nonlinearities g.t / D jt jq sign t , if q  .N C 1/=.N  1/ and N  2 there is no solution with boundary trace given by a Dirac measure. In fact in this case there is no solution with an isolated singularity. In other words, isolated point singularities are removable. (For details see Subsection 3.4.3 and 4.2.1.) The second aspect occurs whenever g satisfies the Keller–Osserman condition discussed below. This condition is satisfied by power nonlinearities for every q > 1 and by the exponential nonlinearity. J.B. Keller [60] and R. Osserman [96] provided a sharp condition on the growth of g at infinity which guarantees that the set of solutions of (4) is uniformly bounded from above in compact subsets of . Qualitatively the condition means that the superlinearity of g at infinity is sufficiently strong. Assuming that this condition holds uniformly with respect to x 2 , they derived an a priori estimate for solutions of (4) in terms of .x/ D dist .x, @/. This estimate implies that equation (4), in bounded domains, possesses a maximal solution. If, in addition,  satisfies the classical Wiener condition then the maximal solution blows up everywhere on the boundary. (If g.x, 0/ D 0 the boundedness assumption on the domain is not needed.) A solution that blows up everywhere on the boundary is called a large solution. Evidently, large solutions do not posses a boundary trace in M.@/.

viii

Preface

In Section 3.1 we show that every positive solution has a boundary trace that is given by an outer regular Borel measure; however this measure need not be finite. If the solution is moderate this reduces to the boundary trace previously mentioned. The boundary trace  N of a positive solution u has a singular set F (possibly empty) such that  N is infinite on F while N is a Radon measure on @ n F . The singular set is closed. A point y 2 @ is singular (relative to u) if y 2 F and regular otherwise. The singular and regular boundary points are determined by a local integral condition. A boundary trace N can also be represented by a couple .F , / where F is the singular set of the trace and  is a Radon measure on @ n F . The set of all positive measures that can be represented in this manner is denoted by Breg . A solution whose boundary trace is of the form .F , 0/ is called a purely singular solution. Assumimg that the Keller–Osserman condition holds uniformly in , for every compact set F  @ there exists a solution UF that is maximal in the set of solutions vanishing on @ n F (see Section 3.2). UF is called the maximal solution relative to F . In the subcritical case, the boundary trace of UF is .F , 0/. In the supercritical case, the singular set of UF – denoted by kg .F / – may be smaller than F . The maximal solutions UF play a crucial role in the study of the boundary value problem u C g.x, u/ D 0 in  uD N on @

(5)

when g 2 G0 and N 2 Breg. In Section 3.3 we present a general result providing necessary and sufficient conditions for existence and uniqueness of solutions of (5) assuming that g satisfies the local Keller–Osserman condition and the global barrier condition and that, for every x 2 , g.x, / is convex. (See definitions 3.1.9 and 3.1.10.) These conditions are sufficient for the existence of the maximal solution UF . In Section 3.4 we study problem (5) when g is given by g.x, t / D .x/ˇ jt jq1 t ,

q > 1, ˇ > 2.

(6)

Assuming that  is a smooth domain we show: (i) A g-barrier exists at every boundary point and the global barrier condition holds and (ii) g is subcritical if and only if 1 < q < qc .˛/ :D .N C ˇ C 1/=.N  1/. Next we apply the result of Section 3.3 to problem (5) with g as above assuming that q is in the subcritical range. We show that, under these assumptions: Problem (5) possesses a unique solution for every N 2 Breg. There follows a description of the main steps in the proof of this result: I.

For every y 2 @ there is a unique solution with boundary trace .¹yº, 0/ denoted by u1,y .

ix

Preface

II. If u is a solution with singular boundary set F then for every y 2 F , u  u1,y . Using these two results we show that: III. For every compact F  @, the maximal solution UF is the unique solution with trace .F , 0/. The proof is completed by establishing the following: IV. If, for every compact set F  @, (5) has a unique solution with boundary trace .F , 0/ then the boundary value problem has a unique solution for every measure N 2 Breg . Two particular cases of the boundary value problem (5) have received special attention in the literature. The first is the case of large solutions already mentioned above. In the language of boundary traces, the singular boundary set of a large solution is the whole boundary. In the case of Lipschitz domains, the global Keller–Osserman condition implies the existence of a large solution. However, in more general domains, the maximal solution may not blow up everywhere on the boundary. Therefore, in such a case a large solution does not exist. The question of existence and uniqueness of a large solution under various assumptions on g and  has been a subject of intense study. In addition to its intrinsic interest, this topic is useful in delineating the limitations that are naturally imposed on the goals of our study of general boundary value problems. The subject of large solutions is discussed in detail in Chapter 5. The second case to receive special attention is that of solutions with isolated singularities. If the nonlinearity is subcritical then, for every y 2 @ there exist moderate solutions with isolated singularity at y. If, in addition, a g-barrier exists at y then there exist non-moderate solutions with an isolated singularity at y. Such a solution is called a ‘very singular solution’. Alternatively we say that the solution has a ‘strong isolated singularity’ at y. Assume that g is subcritical and that a g-barrier exists at y 2 @. Let uk,y denote the solution with boundary trace kıy . For k > 0 this solution is dominated by kP ., y/ (where P denotes the Poisson kernel); therefore it is a moderate solution. However, the existence of a barrier at y implies that u1,y D lim uk,y k!1

(7)

is a solution of the equation which vanishes on @ n ¹yº. Evidently this solution has a strong singularity at y. The analysis of the set of solutions with strong isolated singularities plays an important role in the study of boundary value problems in the subcritical case. A question of special interest is the uniqueness of the very singular solution at

x

Preface

y. A related question is that of the asymptotic behavior of such solutions. These questions are studied in Section 3.4 and Chapter 4 for various families of nonlinearities, applying different methods. Several other problems associated to singular and large solutions are considered in Chapter 6. These include: the limit of fundamental solutions when the mass goes to infinity; symmetry of large solutions; higher order terms in the asymptotics of large solutions and their dependence on the geometry of the domain. This monograph was conceived and planned jointly by the two authors. However, it falls into two essentially independent parts. The first part, consisting of Chapters 1–3, was written by the first author and is an outgrowth of a set of notes [73] originally intended for inclusion in a handbook planned by North Holland Ltd. (The handbook project was terminated before the completion of the notes.) The second part, consisting of Chapters 4–6, was written by the second author. The authors are grateful to Dr. Mousomi Bhakta and Dr. Nguyen-Phuoc Tai for carefully reading the manuscript and for suggestions that contributed to the improvement of the presentation.

Contents

Preface

v

1

Linear second order elliptic equations with measure data

1

1.1 Linear boundary value problems with L1 data . . . . . . . . . . . . . . . . . . .

1

1.2 Measure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3 M-boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 The Herglotz–Doob theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Subsolutions, supersolutions and Kato’s inequality . . . . . . . . . . . . . . . 20 1.6 Boundary Harnack principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2

Nonlinear second order elliptic equations with measure data

33

1

2.1 Semilinear problems with L data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Semilinear problems with bounded measure data . . . . . . . . . . . . . . . . . 36 2.3 Subcritical nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Weak Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2.3.2 Continuity of G and P relative to Lw norm . . . . . . . . . . . . . . . 2.3.3 Continuity of a superposition operator . . . . . . . . . . . . . . . . . . . g 2.3.4 Weak continuity of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g 2.3.5 Weak continuity of S@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44 47 48 52 56

2.4 The structure of Mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5 Remarks on unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3

The boundary trace and associated boundary value problems

66

3.1 The boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Moderate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 70 78

3.2 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 The boundary value problem with rough trace . . . . . . . . . . . . . . . . . . . 81

xii

Contents

3.4 A problem with fading absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The similarity transformation and an extension of the Keller–Osserman estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Barriers and maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The critical exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 The very singular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88 89 94 96

3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4

Isolated singularities

108

4.1 Universal upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.1.1 The Keller–Osserman estimates . . . . . . . . . . . . . . . . . . . . . . . . 108 4.1.2 Applications to model cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Removable singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Isolated positive singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Isolated signed singularities . . . . . . . . . . . . . . . . . . . . . . . . . . .

114 114 116 124

4.3 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The half-space case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The case of a general domain . . . . . . . . . . . . . . . . . . . . . . . . . .

130 130 131 138

4.4 Boundary singularities with fading absorption . . . . . . . . . . . . . . . . . . . 147 4.4.1 Power-type degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.4.2 A strongly fading absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.5.1 General results of isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.5.2 Isolated singularities of supersolutions . . . . . . . . . . . . . . . . . . 157 4.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5

Classical theory of maximal and large solutions

162

5.1 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.1.1 Global conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.1.2 Local conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2 Large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2.1 General nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2.2 The power and exponential cases . . . . . . . . . . . . . . . . . . . . . . . 171 5.3 Uniqueness of large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3.1 General uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3.2 Applications to power and exponential types of nonlinearities 182

Contents

xiii

5.4 Equations with a forcing term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.4.1 Maximal and minimal large solutions . . . . . . . . . . . . . . . . . . . . 184 5.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6

Further results on singularities and large solutions

195

6.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.1.1 Internal singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.1.2 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.2 Symmetries of large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.3 Sharp blow up rate of large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.3.1 Estimates in an annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.3.2 Curvature secondary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.4 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Bibliography

239

Index

247

Chapter 1

Linear second order elliptic equations with measure data 1.1 Linear boundary value problems with L1 data We begin with linear boundary value problems with L1 data of the form u D f uD

in , on @

(1.1.1)

where  is a domain in RN . To simplify the presentation we shall assume that N  3. However, with slight modifications, most of the results apply as well to N D 2. Unless otherwise stated, we assume that  is a bounded domain of class C 2 . Definition 1.1.1. A bounded domain   RN is of class C 2 if there exists a positive number r0 such that, for every X 2 @, there exists a set of Cartesian coordinates  D  X , centered at X, and a function FX 2 C 2.RN 1 / such that FX .0/ D 0, rFX .0/ D 0 and  \ Br0 .X/ D ¹ : jj < r0, 1 > FX .2 , : : : , N /º.

(1.1.2)

The set of coordinates  X is called a normal set of coordinates at X and FX is called the local defining function at X. The normal set of coordinates at X is not uniquely defined. However, the direction of the positive 1X axis coincides with the direction of the unit normal at X pointing into the domain and two sets of normal coordinates at X are related by a rotation around the 1X axis. As  is bounded, @ can be covered by a finite number of balls ¹Br0 .Xi /ºkiD1, X1, : : : , Xk 2 @. Therefore, k@kC 2 :D sup¹kFX kC 2 .BN r

0 .0//

: x0 2 @º < 1

and there exists 2 C.0, 1/ such that D 2FX has modulus of continuity for every X 2 @. The pair .r0 , k@kC 2 / is called a C 2 characteristic of . We denote by G and P the Green and Poisson kernels respectively of  in . N and  2 C.@/ then a classical result states that the boundary value If f 2 C 1 ./ N The solution u is given problem (1.1.1) possesses a unique solution in C 2 ./ \ C./. by Z Z u.x/ D



G.x, y/f .y/dy C

@

P .x, y/.y/dSy .

(1.1.3)

2

Chapter 1 Linear second order elliptic equations with measure data

Put

N :D ¹ 2 C 2 ./ N : D 0 on @º. C02./

N and  2 C 1 .@/ and let 2 C 2 ./. N Multiplying the Assume that f 2 C 1 ./ 0 equation in (1.1.1) by and integrating by parts we obtain Z Z Z f dx D  u dx C @n dS (1.1.4) 



@

where dS denotes the surface element on @ and @n denotes differentiation in the outer normal direction on @. Denote by L1.; / the weighted Lebesgue space with weight , where is a positive measurable function in . Let  be the function given by ´ dist .x, @/ 8x 2  .x/ D 0 otherwise. We note that the integral on the left-hand side of (1.1.4) is well defined for every f 2 L1.; / and the last integral on the right-hand side is well defined for every  2 L1.@/. Accordingly we define a weak solution of (1.1.1) with L1 data as follows: Definition 1.1.2. Assume that f 2 L1.; /,

 2 L1 .@/.

(1.1.5)

A function u 2 L1./ is a weak solution of (1.1.1) if it satisfies (1.1.4) for every N

2 C02./. Recall the following estimates for G and P (see e.g. [53]): ˇ ˇ ˇ ˇ G.x, y/  min..x/, .y// jx  yj1N , ˇrxk G.x, y/ˇ  C jx  yj2kN , (1.1.6) for every x, y 2  and ˇ ˇ ˇ ˇ P .x, y/  .x/ jx  yjN , ˇrxk P .x, y/ˇ  C jx  yj1kN (1.1.7) for every x 2 , y 2 @ and k D 0, 1, 2, : : : . If f , h are non-negative functions on a domain D, the notation h  f means 9c > 0 such that

c 1 h  f  ch.

In (1.1.6) and (1.1.7) the constants depend only on the C 2 characteristic of the boundary and the diameter of the domain. Employing these estimates one can establish the existence and uniqueness of weak solutions.

3

Section 1.2 Measure data

Proposition 1.1.3. For every f 2 L1 .; / and  2 L1.@/ problem (1.1.1) possesses a unique weak solution. The solution is given by (1.1.3) and satisfies,   (1.1.8) kukL1 ./  C kf kL1 .;/ C kkL1 .@/ and

  kuC kL1./  C kfCkL1 .;/ C kCkL1 .@/

(1.1.9)

where C is a constant depending only on . Proof. First observe that, by virtue of estimates (1.1.6) and (1.1.7), the function u defined by (1.1.3) is in L1 ./ and satisfies (1.1.8). Approximate f and  by sequences ¹fn º  Cc1 ./ and ¹nº 2 Cc1 .@/ in 1 L .; / and L1.@/ respectively. Denote by un the solution of (1.1.1) with f ,  replaced by fn , n. By (1.1.8) ¹un º converges in L1 ./ to u. Furthermore, as un satisfies Z Z Z un  dx D fn dx  n@n dS,  



@

we conclude that u satisfies (1.1.4). If f ,  are non-negative then they can be approximated by sequences ¹fn º  Cc1./ and ¹nº 2 Cc1 .@/ consisting of non-negative functions. By the maximum principle, un  0 and consequently u  0. In the general case let v1 and v2 be the weak solutions of (1.1.1) with f ,  replaced by fC, C and f ,  respectively, where fC D max.f , 0/,

f D max.f , 0/.

Then vi  0 and u D v1  v2 . Therefore uC  v1 and (1.1.9) follows from (1.1.8)  applied to v1 .

1.2 Measure data The previous result can be fully extended to the case where the functions f ,  are replaced by measures. Recall that a positive Borel measure on  is called a Radon measure if it is bounded on compact sets. A Borel measure with possibly changing signs is called a signed Radon measure if it is the difference of two positive Radon measures, at least one of which is finite. If  is a signed Radon measure, denote by C and  its positive and negative parts and by jj the total variation measure jj D C C  . Denote by M./ the space of finite Borel measures endowed with the norm kkM./ D jj./

(1.2.1)

4

Chapter 1 Linear second order elliptic equations with measure data

and by M ./ (or M.; /) the space of signed Radon measures  such that Z  d jj < 1. (1.2.2) kkM ./ :D 

Finally denote by Mloc ./ the space of set functions  on Bc ./ D ¹E b  : E Borelº such that 1K is a finite measure for every compact K  . Following Bourbaki, such a set function is called a real valued Radon measure. A set function  belongs to this space if and only if it is the difference of two positive Radon measures 1, 2 . If at least one of these two is finite then  is a signed Radon measure. However, if both are unbounded then  is not a measure on . The space Mloc./ can be characterized as the set of continuous linear functionals on Cc ./ endowed with the inductive limit. A functional ` on Cc ./ is continuous in this sense if and only if, for every compact set K  , ` is continuous on CK D ¹f 2 Cc ./ : supp f  Kº. Definition 1.2.1. Assume that  2 M ./,

 2 M.@/.

(1.2.3)

A function u 2 L1./ is a weak solution of the problem u D  in , u D  on @ if it satisfies

Z

Z 

d D 

(1.2.4)

Z 

u dx C

@

@n d

(1.2.5)

N for every 2 C02 ./. Theorem 1.2.2. Assume  2 M ./ and  2 M.@/. (i) Problem (1.2.4) has a unique weak solution u given by Z Z G.x, y/d.y/ C P .x, y/d.y/. u.x/ D 

(1.2.6)

@

Furthermore

  kukLp ./  C.p/ kkM ./ C kk M.@/ ,

1p<

N , N 1

(1.2.7)

and

  kuC kLp ./  C.p/ kC kM ./ C kC kM.@/ ,

where C.p/ is a constant depending only on p and .

1p<

N , (1.2.8) N 1

5

Section 1.2 Measure data

(ii) For every p 2 Œ1, NN1 /, u 2 Wloc ./ and, if 0 b ,   kukW 1,p .0 /  C.p, 0 / kkM.0 / C kkM.@/ . 1,p

(1.2.9)

(iii) If  D 0 then (1.2.7) and (1.2.8) hold for every p 2 Œ1, N=.N 2/. If, in addition,  2 M./ then, for every p 2 Œ1, NN1 /, u 2 W 1,p ./ and kukW 1,p ./  C.p/ kkM./ ,

(1.2.10)

where C.p/ is a constant depending only on p and . Proof. The uniqueness of solutions of the classical Dirichlet problem implies that problem (1.2.4) has at most one solution. Let u be the function defined by (1.2.6). In the first part of the proof we show that this function satisfies estimates (1.2.7)–(1.2.10); in the second part we show that u is the weak solution of (1.2.4). If  D 0 and  2 M., / then, by (1.1.6), Z jx  yj2N .y/d jj .y/. (1.2.11) ju.x/j  C 

Let ˛ denote the function ˛ .x/ D jxj˛N . The measure  jj is bounded with compact support in RN and 2 2 Lp .RN / for 1  p < N=.N  2/. Hence 2  loc .jj/ 2 Lp .RN / and (1.2.7) holds. If  D 0,  2 M.@/ then, by (1.1.7), Z jx  yj1N d jj .y/. (1.2.12) ju.x/j  C @

Hence, for 1  p < N=.N  1/, Z Z 1=p jx  yjp.1N / dx d jj .y/  C.p/ kkM.@/ . kukLp ./  C @



This completes the proof of (1.2.7). Estimate (1.2.8) follows by the same argument as in the proof of Proposition 1.1.3. If  D 0 and  2 M./ then, by the second inequality in (1.1.6), Z jru.x/j  C jx  yj1N d jj .y/. 

Therefore, for 1  p < N=.N  1/, u 2 W 1,p ./ and (1.2.10) holds. Next we verify (1.2.9). Put Z Z G.x, y/d.y/, u2 D P .x, y/d.y/. u1 .x/ D 

@

6

Chapter 1 Linear second order elliptic equations with measure data

If 0 b  then 10 2 M.0 / so that (1.2.10) implies that u1 satisfies (1.2.9). On the other hand u2 is harmonic in  and for every compact subdomain 0   we have sup ju2 j  C dist .0 , @/1N jj. 0

This proves (1.2.9). We turn to the second part of the proof: to show that u is a weak solution of (1.2.4). First we prove this statement in the case that  D 0 and  2 M./. Let ¹fnº be a sequence of functions in Cc1 ./ such that fn *  weakly relative N Denote by un the solution of (1.2.4) with  replaced by fn and  D 0. In to C0 ./. this case we know that Z un D G.x, y/fn .x/dx. 

By (1.2.10), ¹un º is bounded in W 1,p ./, 1  p < N=.N 1/. Therefore there exists a subsequence ¹unk º which converges in Lp ./. Since Z Z unk  dx D fnk dx  

for every 2

N C02 ./

for every 2

N C02 ./.



we conclude that w D lim unk satisfies Z Z  w dx D

d 



It follows that w is the unique solution of the problem w D  in ,

w D 0 on @.

Since the limit does not depend on the subsequence it follows that w D lim un . Next we show that w D u in the case when  2 M./ and   0. In this case we may choose the sequence ¹fn º so that fn  0. Given > 0, let ' 2 Cc1 .RN / be a function such that 0  '  1, Then

' D 1 in RN n B .0/.

' D 0 in B=2 .0/,

Z

un .x/ D D

Z 

G.x, y/fn .y/dy G.x, y/' .jx  yj/fn .y/dy C

Z 

G.x, y/.1  ' .jx  yj//fn .y/dy

D: un,1 .x/ C un,2 .x/. N therefore, For every x 2  the function y 7! G.x, y/' .jx  yj/ is continuous in ; the weak convergence of ¹fnº implies Z G.x, y/' .jx  yj/d.y/ 8x 2 . un,1 .x/ ! 

7

Section 1.2 Measure data

Thus

Z w.x/  u.x/ D



G.x, y/.1  ' .jx  yj//d.y/ C lim un,2 .x/. n!1

Let F be a compact subset of , < 14 dist .F , @/ and .F / :D ¹x 2 RN : dist .x, F / < º. Then

Z Z

Z F

un,2 dx D 

Z 

G.x, y/.1  ' .jx  yj//dxfn .y/dy Z fn dy sup G.x, y/dx. F

jxyj<

y2.F /

Z

Hence

Z

lim sup n!1

F

un,2 dx  ./ sup y2.F /

jxyj<

G.x, y/dx

and the last term tends to zero as ! 0. Similarly we obtain, Z Z lim G.x, y/.1  ' .jx  yj//d.y/ D 0. !0 F



Consequently (using the lemma of Fatou): Z .w  u/dx 0 Z ZF Z G.x, y/.1  ' .jx  yj//d.y/ C lim inf un,2 .x/dx  F

n!1



F

and the right-hand side tends to zero as ! 0. It follows that u D w in F and (as F is an arbitrary compact subset of ) u D w in . Now we consider the case when  2 M ./ and   0. We approximate  by a sequence ¹k º of smooth domains such that k "  and put k D 1k . Let vk be the solution of v D k in , v D 0 on @. By the previous part of the proof Z vk D



G.x, y/dk .y/

and vk " u. Since vk satisfies Z 

Z

dk D 



vk  dx,

8

Chapter 1 Linear second order elliptic equations with measure data

N and u 2 L1./ we conclude that for every 2 C02 ./ Z Z

d D  u dx 



for every as above. Thus u is the weak solution of (1.2.4). To show that this result remains valid when  is not necessarily positive we apply the last statement to C and  separately. Finally we prove that Z u.x/ D

P .x, y/d.y/ @

is a weak solution of (1.2.4) with  D 0. Let ¹hnº be a sequence of smooth functions converging weakly to  relative to C.@/. Then Z wn .x/ D P .x, y/hn dSy @

is the classical solution of w D 0 in ,

w D hn on @. Z

Z

Thus 0D



wn  dx 

@

hn @n dS

(1.2.13)

N As P .x, / 2 C.@/ for every x 2 , it follows that wn ! u for every 2 C02 ./. everywhere in . By (1.2.7) ¹wn º is bounded in Lp ./ for some p > 1. These two facts imply that wn ! u in L1 ./. Therefore (1.2.13) implies Z Z u dx  @n d 0D 

@

N C02 ./.

for every 2 Thus u is a weak solution of (1.2.4) if either  D 0 or  D 0. By linearity this implies the result in the general case.  We mention the following useful corollary. Corollary 1.2.3. Let  be a real valued Radon measure in  and suppose that u 2 L1loc./ satisfies u D  in , i.e., Z Z u dx D

d 8 2 Cc1 ./. (1.2.14)  D

D

1,p

Then, u 2 Wloc ./ for every p 2 Œ1, N=.N  1// and, for every domain D b  of class C 2 , Z Z Z u dx D

d  ub@D @n dS, (1.2.15)  for every 2

N C02 .D/.

D

D

@D

1

Here ub@D denotes the L Sobolev trace of u on @D.

9

Section 1.2 Measure data

Z

Proof. Put vD .x/ D

D

G.x, y/ d.y/ 8x 2 D.

By Theorem 1.2.2, applied to the measure D :D 1D in , Z Z N ' dD D vD ' dx 8' 2 C02./ 



and vD 2 Wloc1,p ./. Thus vD 2 W 1,p .D/,

vD D  in D

for every p 2 Œ1, N=.N 1//. It follows that uvD is harmonic in D and consequently, u 2 Wloc1,p .D/. As D is any C 2 domain strongly contained in , it follows that u 2 1,p Wloc ./. Consequently, by the Sobolev trace theorem, u possesses an L1 trace on every compact N  1-dimensional C 1 manifold contained in . N @/ and let 2 Cc1 ./ be a function such that 0   1 and Let 0 :D dist .D, ´ 1 if dist .x, D/ < 0=2 .x/ D 0 if dist .x, D/ > 3 0=4. For 0 < < 0 =2 let u :D J .u / and  D J . /. Then u D  in D (in N Therefore, the classical sense) and u 2 C.D/. Z Z Z  u  dx D

 dx  u @n dS, D

for every 2

N C02 .D/.

D

@D

Letting ! 0 we obtain (1.2.15).



Remark 1.2.2.A. Let ¹n º  M ./, ¹n º  M.@/ and assume that n !  strongly in M ./ and n !  strongly in M.@/. Let u (resp. un ) be the weak solution of (1.2.4) with data ,  (resp. with data n , n ). By Theorem 1.2.2 (i), un ! u strongly in Lp ./ for p 2 Œ1, N=.N  1//. Briefly: The L1 weak solution of (1.2.4) is stable with respect to strong convergence of the data. The next result shows that the solution is also stable (in a weaker sense) with respect to an appropriate type of weak convergence of the data. First recall the standard definition of ‘weak convergence’ in M.K/, the space of finite Borel measures on a compact set K  RN . We say that the sequence ¹k º converges ‘weakly’ to  if Z Z f dk ! f d 8f 2 C.K/. K

K

This is in fact weak convergence in the dual, M.K/, of C.K/. The topology of weak convergence is metrizable, a bounded sequence is pre-compact, i.e. contains a weakly

10

Chapter 1 Linear second order elliptic equations with measure data

convergent subsequence and every weakly convergent sequence is bounded. For this and other properties of weak convergence of measures we refer the reader to any standard measure theory textbook. When  is a bounded domain, M./ is the dual of N D ¹f 2 C./; N f D 0 on @º. C0./ N is the closure of Cc ./ in C./. N In this case we say that ¹k º conNote that C0./ verges ‘weakly’ to  if Z Z N f dk ! f d 8f 2 C0./. 



As before, the topology of weak convergence is metrizable and the properties mentioned above persist. Finally, consider the space M ./ when  is a bounded C 1 domain. This space is the dual of N / D ¹h : h= 2 C0./º. N C0 .; N means that h= has a continuous extension to , N which is zero on Here h= 2 C0./ @. Therefore we define: A sequence ¹k º  M ./ converges weakly to  2 M ./ if Z Z N /. f dk ! f d 8f 2 C0 .; 

(1.2.16)



Thus the weak convergence in the sense of (1.2.16) is equivalent to the weak converN Again, the topology of weak gence n *  in M./, i.e. with respect to C0./. convergence is metrizable, a bounded sequence is pre-compact and every weakly convergent sequence is bounded. Definition 1.2.4. A sequence ¹n º  M./ is tight if for every > 0 there exists a neighborhood U of @ such that jn j.U \ / < . Similarly, a sequence ¹n º  M.; / is tight in this space if ¹nº is tight in M./. Remark. If a sequence in M./ is weakly convergent but not tight, it might have a N that is different from the weak limit in M./. Here is a simple weak limit in M./ example. Let ¹An º be a sequence of points in  such that An ! A 2 @. Denote by n (resp. ) the Dirac measure of mass 1 concentrated at An (resp. A). Then, in N ¹n º converges weakly to  but, in M./, it converges weakly to 0. Evidently M./, this sequence is not tight in M./. Theorem 1.2.5. (i) Let ¹n º  M./ and ¹n º  M.@/. Assume that n *  N while n *  relative to C.@/. Let u be the weak solution relative to C0./

11

Section 1.2 Measure data

of (1.2.4) and let un be the weak solution of (1.2.4) with ,  replaced by n , n . Then: 1,p

un * u weakly in Wloc ./,

un ! u strongly in Lp ./,

(1.2.17)

un ! u strongly in Lq ./,

(1.2.18)

for every p 2 Œ1, N=.N  1/. If in addition n D 0 for all n then un ! u strongly in W 1,p ./,

for every p 2 Œ1, N=.N  1/, q 2 Œ1, N=.N  2//. (ii) Let ¹n º be a bounded and tight sequence in M.; / such that n *  relative N /. Let ¹n º  M.@/ and assume that n *  relative to C.@/. Then to C0 .; (1.2.17) holds. Remark 1.1. Note that in part (i) we do not assume ‘tightness’ but in part (ii) this assumption is needed. The following example shows that the conclusion of Theorem 1.2.5 (ii) may fail in the absence of tightness. Let ¹An º be a sequence of points in  converging to a point A 2 @. Put n D a1n ıAn where an D dist .An , @/. (ıA denotes the Dirac measure of mass 1 concentrated at A.) Then ¹n º is bounded in M.; / but it is not tight. Furthermore n * 0 weakly in M.; /. But, if un is the solution of (1.2.4) with boundary data 0 then un ! P ., A/ pointwise in . Note also that every bounded sequence in M./ is tight in M.; / although it may not be tight in M./. Proof. (i) By Theorem 1.2.2 (ii), for every 0 b , ¹un º is bounded in W 1,p .0 /, for every p 2 Œ1, N=.N  1//. Consequently there exists a subsequence ¹unk º and 1,p v 2 Wloc ./ such that unk * v

1,p

weakly in Wloc ./

for all p as above. By the Sobolev imbedding theorem unk ! v

in Lqloc./,

1  q < N=.N  2/.

By taking a further subsequence we may assume that unk ! v a.e. in . By (1.2.7) ¹un º is uniformly bounded in Lp ./, 1  p < N=.N  1/. Therefore ¹un º is uniformly integrable in Lr ./, 1  r < N=.N  1/. Since unk ! v a.e. in  we conclude that unk ! v

in Lp ./,

1  p < N=.N  1/.

Now, for every n, Z Z Z un ' dx D ' dn   



@

@n' dn

N 8' 2 C02 ./.

12

Chapter 1 Linear second order elliptic equations with measure data

Replacing n by nk and taking the limit as k ! 1 we obtain Z Z Z N  v' dx D ' d  @n ' d 8' 2 C02./. 



@

Thus v is the weak solution of (1.2.5) and, by uniqueness, v D u. Since the limit does not depend on the subsequence we obtain (1.2.17). If in addition, n D 0 for all n, (1.2.18) is obtained by the same argument, using Theorem 1.2.2 (iii). (ii) Let k be a function in Cc1 ./ such that 0  k  1, and ´ 1 if .x/ > 2k k .x/ D 0 if .x/ < 2k1. Note that k " 1 in . Let u0k,n (resp. u0k ) denote the weak solution of (1.2.4) with  replaced by n (resp. by  k ). Put vk,n D un  u0k,n , Thus

k

vk D u  u0k .

Z vk,n.x/ D

and



G.x, y/.1 

k /dn .y/

G.x, y/.1 

k /d.y/.

Z vk .x/ D



The tightness assumption implies that lim kn .1 

k!1

k /kM.;/

D0

(1.2.19)

uniformly with respect to n. Therefore, by Theorem 1.2.2, kv k p ! 0 as k ! 1,  k L ./ vk,n  p ! 0 as k ! 1, L ./

uniformly with respect to n,

for 1  p < N=.N  1//. For fixed k, ¹ k n º converges strongly to u0k,n * u0k weakly in Wloc ./, 1,p

k .

(1.2.20)

Therefore, by part (i),

u0k,n ! u0k strongly in Lp ./,

for every p 2 Œ1, N=.N  1//. Combining (1.2.20) and (1.2.21) we obtain (1.2.17).

(1.2.21) 

13

Section 1.3 M-boundary trace

1.3 M-boundary trace Definition 1.3.1. A sequence ¹Dn º is an exhaustion of  if DN n  DnC1 and Dn " . We say that an exhaustion ¹Dnº is of class C ˛ if each domain Dn is of this class. If, in addition,  is a C ˛ domain, ˛ > 0, and the sequence of domains ¹Dnº is uniformly of class C ˛ we say that ¹Dnº is a uniform C ˛ exhaustion. Note. ¹Dnº is uniformly of class C ˛ if there exists r0 , 0, n0 such that, for every X 2 @D: There exists a system of Cartesian coordinates  centered at X, a sequence ¹fn º  ˛ C .BrN0 1 .0// and f 2 C ˛ .BrN0 1 .0// such that the following statement holds. Let Q0 :D ¹ D .1 ,  0 / 2 R RN 1 : j 0 j < r0, jN j < 0º. Then the surfaces @Dn \ Q0 , n > n0 and @ \ Q0 can be represented by 1 D fn . 0 / and 1 D f . 0 / respectively and fn ! f

in C ˛ .BrN0 1 .0//.

At this point we introduce some additional notation and a few related technical remarks. Recall our basic assumption:  is a bounded domain in RN whose boundary † is a 2 C manifold. We use the notation: .x/ D dist .x, @/,

†ˇ D ¹x 2  : .x/ D ˇº, Dˇ D ¹x 2  : .x/ > ˇº, ˇ D  n DN ˇ .

(1.3.1)

The outward, unit normal vector to @ at x0 is denoted by nx0 . Proposition 1.3.2. There exists a positive number ˇ0 such that: N ˇ , there exists a unique point .x/ 2 @ such that (a) For every point x 2  0 jx  .x/j D .x/. This implies, x D .x/  .x/n.x/ . N ˇ / and C 1 . N ˇ / respec(b) The mappings x 7! .x/ and x ! 7 .x/ belong to C 2 . 0 0 tively. Furthermore, lim r.x/ D n.x/ . x!.x/

N ˇ 7! Œ0, ˇ0  † the mapping given by ….x/ D ..x/, .x//. (c) Denote by … :  0 1 Then … is a C - diffeomorphism. For the proof we refer the reader to [53] and [82]. In view of this result, ., / may serve as a set of coordinates in a strip around the boundary. These are called the flow coordinates of . In the following lemmas we state some consequences of the proposition.

14

Chapter 1 Linear second order elliptic equations with measure data

Lemma 1.3.3. Let ¹Dnº be a uniform C 2 exhaustion of  and let ´ 8x 2 Dn dist .x, @Dn / n .x/ D . N n Dn 0 8x 2  Then there exists a positive number d0 such that n !  n ! 

N d /, in C. 0 N d n ˇ / in C 2 . 0

8ˇ 2 .0, d0/,

sup knkC 2 .N d / < 1. 0

Lemma 1.3.4. Let ¹ nº be a strictly decreasing sequence converging to zero. If 1 < ˇ0 the sequence of domains ¹Dn º is a uniform C 2 exhaustion. Lemma 1.3.5. Let  be a domain of class C 2 and let ¹Dnº be a uniform C 2 exhausN there exists a sequence ¹'n º and a function ' such tion. Then, for every h 2 C 2 ./, that 'n 2 C02 .DN n /, @n 'n b@Dn D h on @Dn, N ' 2 C02./, @n 'b@ D h on @, k'n kC 2 .DN n / < c khkC 2 ./ N ,

'n  cn h N 0 /. The constant c is indepenand, for every domain  b , 'n =n ! '= in C 2 . dent of h and n, but depends on the exhaustion. 0

Proof of Lemma 1.3.5. Let n and d0 be as in Lemma 1.3.3. The functions 'n can be constructed in such a way that 'n .x/ D n .x/h.x/ for x 2 Dn such that n.x/ <  d0=2. Notation. Integrating on an .N  1/-dimensional C 1 manifold, we write dS D d HN 1 where HN 1 denotes Hausdorff measure. 1,p

Definition 1.3.6. Let u 2 Wloc ./ for some p > 1. We say that u possesses an M-boundary trace on @ if there exists  2 M.@/ such that, for every uniform C 2 N exhaustion ¹n º and every h 2 C./, Z Z ub@n hdS ! h d. (1.3.2) @n

@

Here ub@n denotes the Sobolev trace. The M-boundary trace of u is denoted by tr u. Let A be a relatively open subset of @. We say that u has M-boundary trace  on A if  is either a finite measure or a positive Radon measure on A and, for every N vanishing in a neighborhood of @ n A, the function u has M-boundary

2 C 2./ trace  (defined as zero on @ n A). The M-boundary trace of u on A is denoted by trA u.

15

Section 1.3 M-boundary trace

N then it also holds Remark 1.3.6.a. If u  0 and (1.3.2) holds for every h 2R C 2 ./ N for every h 2 C./. Indeed, taking h D 1 we find that ¹ @n uº is bounded and if N and kf  hkC./ N h 2 C 2./ f 2 C./, N < then Z lim sup @n

ub@n jh  f jdS < C .

N we obtain (1.3.2) N by a sequence ¹hk º  C 2 ./ Therefore approximating h 2 C./ N for every h 2 C./. Remark 1.3.6.b. If u 2 W 1,p ./ for some p > 1 then the Sobolev trace = Mboundary trace. Proposition 1.3.7. Let  2 M ./ and  2 M.@/. Then a function u 2 L1./ satisfies (1.2.5) if and only if (i) (ii)

u D  tr u D 

in  (in the sense of distributions), on @ (in the sense of Definition 1.3.6).

(1.3.3)

Proof. Assume that u satisfies (1.2.5). Then u D  in the sense of distributions and Corollary 1.2.3 applies. Assertion 1.3.7.a. If

Z v.x/ :D

G.x, y/ d.y/ 

8x 2 

then v possesses an M-boundary trace on @ and tr@ v D 0. If  2 M./ then v 2 W01,p ./. Because of linearity it is sufficient to prove the assertion in the case   0. Let ¹k º be a uniform C 2 exhaustion of . Put k .x/ D dist .x@k / for x 2 k and let d0 be a positive number such that k 2 C 2.QN k / where Qk :D ¹x 2 k : k .x/ < d0 º. For every k 2 N, let k be a function in Cc1 ./ such that 0  k  1, ´ 1 if k .x/ > 2k d0 k .x/ D 0 if k .x/ < 2k1 d0 and k D 0 in  n k . Note that k " 1 in . Denote Z Z G.x, y/ k d, wk D G.x, y/.1  vk D 



k / d.

N k , vk D 0 on @ and vk 2 W 1,p .kC1 /. We observe that vk is harmonic in  n  1,p Therefore vk 2 W0 ./. If  2 M./ then vk ! v in W 1,p ./; consequently,

16

Chapter 1 Linear second order elliptic equations with measure data 1,p

in this case, v 2 W0 ./. In the general case, ¹ k º !  strongly in M.; /; therefore vk ! v and wk ! 0 in Lq ./, 1  q < N=.N  1/. Let m be the solution of  D 1 in m ,

D 0 on @m .

Applying (1.2.15) to wk in m , with D m , we obtain Z Z Z wk dx D

m .1  k /d  wk b@m @n m dS. m

m

(1.3.4)

@m

The choice of m implies that there exist positive constants c1 , c2 , independent of m, such that c1  @n m b@m  c2 . Z

Therefore

1 wk b@m dS  c1 @m

Z m

wk dx.

Given > 0, choose k D k. / sufficiently large so that for all m, Z @m

R 

(1.3.5)

wk dx < . It follows that,

wk b@m dS  =c1 .

1,p

Since vk 2 W0 ./,

Z @m

vk b@m dS ! 0 as m ! 1.

In particular, for k D k. / there exists m D m. / such that, Z vk b@m dS  8m  m. /. @m

As v D vk C wk it follows that Z  1 vb@m dS  1 C c1 @m

8m  m. /.

This proves Assertion 1.3.7.a. Assertion 1.3.7.b. Put w.x/ D

Z P .x, y/ d.y/ @

8x 2 .

w possesses an M -trace on @ and tr@ w D . The statement is obvious when  2 C.@/. If  2 L1 .@/ we obtain the result by approximating it in L1 by continuous functions.

17

Section 1.3 M-boundary trace

We turn to the general case. As before, it is sufficient to prove the assertion in the case   0, in which case u  0. We recall that, for non-negative u, it is sufficient to N Let ¹n º be a uniform C 2 exhaustion of . prove (1.3.2) for functions h 2 C 2 ./. 2 N Given h 2 C ./ let ¹'n º and ' be as in Lemma 1.3.5. By Corollary 1.2.3, for k < n, Z Z Z  u'n dx D  ub@n h dS C ub@k h dS. (1.3.6) Nk n n

@n

Since u 2 L1./

@k

Z lim

k!1 nk

udx D 0.

By construction, supN n j'n j is bounded by a constant independent of n. Therefore the left-hand side of (1.3.6) tends to zero as k ! 1. Thus Z In D ub@n h dS n D 1, 2 : : : @k

converges as n ! 1. Appealing again to Corollary 1.2.3, Z  u'n dx D In . n

By Lemma 1.3.5, ¹'n º is bounded and converges pointwise to '. Therefore Z Z Z  u'n dx !  u'dx D hd. n



@

Z

In conclusion, In !

hd @

N In view of Remark 1.3.6.a, this fact implies Assertion 1.3.7.b for every h 2 C 2 ./. for positive  and hence also for signed measures. The two assertions proved above, together with Theorem 1.2.2, imply: if u satisfies (1.2.5) then it satisfies (1.3.3). Conversely, assume that u satisfies (1.3.3). Let h 2 C 2 .@/ and let ¹Dnº, ' and ¹'n º be as in Lemma 1.3.5. Applying Corollary 1.2.3 in Dn we obtain Z Z Z  u'n dx D 'n d  ub@D h dS. (1.3.7) Dn

Dn

@Dn

As ¹'n =º and ¹'n º are bounded sequences converging to '= and ' respectively, it follows that Z Z Z Z u'n dx D u' dx, lim 'n d D ' d. lim Dn



Dn



18

Chapter 1 Linear second order elliptic equations with measure data

By assumption tr@ u D ; therefore, Z Z ub@Dn h dS D lim @Dn

h d. @



Consequently (1.3.7) implies that u satisfies (1.2.5).

Remark 1.2. The proof shows that, if u is a solution of the equation u D  and satisfies (1.3.2) for some uniform C 2 exhaustion then it satisfies (1.3.2) for every such exhaustion.

1.4 The Herglotz–Doob theorem The next theorem is due to Herglotz in the case of positive harmonic functions and to Doob in the case of positive superharmonic functions. Theorem 1.4.1. Suppose that u 2 L1loc ./ is a positive superharmonic function, i.e., u  0 in the sense of distributions. Then u 2 L1./ and there exist measures  2 MC.; / and  2 MC.@/ such that Z Z u.x/ D P .x, y/ d.y/ C G.x, y/ d.y/ 8x 2 . (1.4.1) @



Proof. Since u is a positive distribution it is represented by a positive Radon measure : u D  in . (1.4.2) There exists ˇ0 > 0 such that, if ˇ 2 .0, ˇ0 , the domain Dˇ (see (1.3.1)) is a C 2 domain and ./ 2 C 2 . n Dˇ0 /. Let ¹ˇ : 0 < ˇ < ˇ0 º be a family of non-negative functions such that ˇ 2 C02.DN ˇ /, ˇ D .  ˇ/ in Dˇ n Dˇ0 and   sup ˇ D D ˛ < 1. ˇ

0<ˇ <ˇ0

By Corollary 1.2.3, Z 

Dˇ

Z uˇ dx D

Dˇ

Z ˇ d 

@Dˇ

ub@Dˇ @n ˇ dS,

(1.4.3)

where, in the last integral, ub@Dˇ denotes the L1 Sobolev trace of u on @Dˇ . As the first integral on the right-hand side is non-negative and ˇ is uniformly bounded, we obtain, Z Z C Dˇ

u dx  

@Dˇ

ub@Dˇ @nˇ dS.

(1.4.4)

19

Section 1.4 The Herglotz–Doob theorem

Further, @n ˇ is bounded below by a positive constant c independent of ˇ. Hence, Z Z .C =c/ u dx  ub@Dˇ dS. (1.4.5) Dˇ

@Dˇ

Integrating over an interval .ˇ, / where 0 < ˇ < < ˇ0 we obtain Z Z u dx  .C =c/.  ˇ/ u dx. Dˇ nD

If  ˇ < c=4C then

Z

Z u dx 

Dˇ

D

1 u dx  4

3

Z u dx Dˇ

Z

Z

so that

(1.4.6)

Dˇ

Dˇ nD

u dx  4

u dx. D

Keeping fixed and letting ˇ tend to zero we obtain, Z Z u dx  4 u dx. 3 nD

D

Hence u 2 L1 ./ and, in view of (1.4.5), Z sup ub@Dˇ dS < 1.

(1.4.7)

0<ˇ <ˇ0 Dˇ

It follows that the first and third integrals in (1.4.3) are bounded with respect to ˇ 2 .0, ˇ0 /. Therefore Z sup ˇ d < 1 0<ˇ <ˇ0 Dˇ

and consequently  2 M ./. N and a uniform C 2 exhaustion of , say ¹Dnº, let ¹'n º and ' be Given h 2 C 2./ as in Lemma 1.3.5. In particular, h D @n ' on @,

h D @n'n on @Dn.

Applying once again Corollary 1.2.3 we obtain Z Z Z  u'n dx D 'n d  Dn

Dn

@Dn

hub@Dn dS.

(1.4.8)

The fact that u 2 L1./,  2 M ./ and the convergence properties of ¹'n º described in Lemma 1.3.5 imply that the first two integrals in (1.4.8) converge as

20

Chapter 1 Linear second order elliptic equations with measure data

n ! 1 and therefore the third integral converges. Therefore there exists a measure  2 M.@/ such that Z Z hub@Dn dS ! hd @Dn

@

Z

Z

and 



u' dx D

Z 

' d 

hd

(1.4.9)

@

N Thus  is independent of the choice of the exhaustion ¹Dnº and u for every h 2 C 2./. 1 is an L weak solution of (1.2.4). Finally (1.4.1) follows from Theorem 1.2.2. 

1.5 Subsolutions, supersolutions and Kato’s inequality Definition 1.5.1. An averaging kernel is a non-negative function j 2 C 1 .RN / such R that supp j  B1 .0/ and RN j D 1. For every > 0, put j .x/ D N j.x= / and denote by J the operator f 7! j  f , f 2 L1loc .RN /. J is called a smoothing operator. Definition 1.5.2. A function v 2 L1loc ./ is a subsolution of the equation u D ,  2 Mloc ./, if v   in the distribution sense. Let  2 M ./ and  2 M.@/. A function v 2 L1./ is a subsolution of the problem u D  in  u D  on @ , (1.5.1) Z

if 

Z 

v dx 

Z 

d 

@

@n d,

(1.5.2)

N such that  0. for every 2 C02 ./ A supersolution is defined in the same way with the inequalities inverted. Lemma 1.5.3. If u 2 L1loc ./ is either a subsolution or a supersolution of the equation 1,p u D ,  2 Mloc./, then u 2 Wloc ./ for 1  p < N=.N  1/. In particular, 0 0 2 if  b  and  is of class C then u possesses an M-boundary trace on @0 . Proof. It is sufficient to consider the case where u is a subsolution. In this case u C   0. It follows that there exists a non-negative Radon measure  such that u C  D . 1,p

Consequently u 2 Wloc ./, 1  p < N=.N  1/ and therefore possesses an L1 Sobolev trace on @0 . As remarked before, the Sobolev trace is an M-boundary trace. 

21

Section 1.5 Subsolutions, supersolutions and Kato’s inequality

Proposition 1.5.4. (Kato’s inequality) If u 2 L1loc./ and u  f 2 L1loc./ then .uC /  .sign C u/f

in .

(1.5.3)

in 

(1.5.4)

Consequently, if u 2 L1loc./, .uC /  .sign Cu/u

Proof. Our assumption implies that there exists a non-negative Radon measure  such that u D f  . 1,p

Therefore u 2 Wloc ./ for 1  p < N=.N  1/. Let 0 b  and let ˛ be a non-negative function in Cc1 ./ such that ˛ D 1 in a N 0 . If w :D ˛u then neighborhood of  w  ˛f  2ru  r˛  u˛ D: F 2 L1./. In 0 , w D u and F D f . Therefore it is sufficient to prove (1.5.3) under the assumption that u 2 L1./, f 2 L1 ./ and both u and f vanish outside a compact subset of . First assume that u 2 Cc1 ./. Let h be a convex, non-decreasing function in C 1 .R/ such that h.t / D 1=2 for t  1=4 and h.t / D t for t  1. Given 0 < < 1, put h .t / :D h.t = /. Note that lim .h /0 .t / D sign C .t /,

sup jh0 j D sup jh0 j < 1.

lim h .t / D tC ,

!0

!0

Let  be a non-negative function in Cc1 ./ and put D h0 .u/. Then 2 Cc1 ./ and  0 so that Z Z  Further,

u dx 

f dx. 

Z

Z 





u dx D D 

Z Z





rur dx h0 .u/rur

Z dx C



Z

rh .u/r dx D 

Z

Therefore



jruj2 .h /00 .u/ dx



h .u/ dx.

Z 

h .u/ dx 

f dx 

and, letting ! 0, we obtain Z Z uC  dx  f sign C u dx.  



This proves (1.5.3) in the case that u 2 Cc1 ./.

(1.5.5)

22

Chapter 1 Linear second order elliptic equations with measure data

Now assume only that u 2 L1 ./, f 2 L1 ./ and both vanish outside a compact subset of . We choose for u a representative of its equivalence class which satisfies Z 8 1 < lim u./d  if the limit exists (1.5.6) u.x/ D r !0 jBr .x/j Br .x/ : 0 otherwise. For every real c, put

´

Ec :D ¹x : u.x/ D cº,

wc :D

uc c

in , otherwise.

Let J denote a smoothing operator. Then .J wc /  J f in D and by the first part of the proof, Z Z  .J wc /dx  .J f /sign C.J wc /dx (1.5.7) 



for every non-negative  in Cc1 .D /. Note that .J wc /C ! .wc /C ,

sign C J wc ! sign C wc

a.e. in  n Ec .

More precisely, the convergence holds at every point x 2  such that the limit mentioned in (1.5.6) exists at x and wc .x/ ¤ 0. Of course J f ! f in L1./. Therefore if jEc j D 0 (jEj denotes Lebesgue measure of E) then (1.5.5) holds for wc . Since jEc j D 0 for a.e. real c we obtain Z Z f sign C .u  c/ dx  .u  c/C  dx a.e. c. (1.5.8) Œu>c

Œu>c

Pick a sequence of positive numbers ¹cn º such that cn # 0 and (1.5.8) holds for c D cn for every n. Applying (1.5.8) to c D cn and taking the limit we obtain (1.5.5).  Corollary 1.5.5. If u 2 L1loc./ and u D f 2 L1loc ./ then juj  .sign u/ u in . Proof. Apply (1.5.3) to uC and u .

(1.5.9) 

Corollary 1.5.6. If f 2 L1loc ./, v 2 L1loc./ is a subsolution and w 2 L1loc ./ is a supersolution of the equation u D f then .v  w/C is a subharmonic function. Proof. Since .v  w/  0 this statement is an immediate consequence of Kato’s inequality.  Remark. Assume that v is a subsolution and w is a solution of u D f . Since max.v, w/ D w C .v  w/C, the previous corollary implies that max.v, w/ is a subsolution of this equation. In fact this result holds even if both v and w are subsolutions. This is well known with regard to classical solutions (when f 2 C 1 ./) and it follows in the general case by using the smoothing operator.

23

Section 1.5 Subsolutions, supersolutions and Kato’s inequality

Corollary 1.5.7. Let  2 Mloc./ and f 2 L1loc ./. If u 2 L1loc./ satisfies u   C f

in 

(1.5.10)

then .uC /  C C f sign C .u/

(1.5.11)

juj  C C f sign .u/

(1.5.12)

and In particular, .uC / 2 Mloc ./ and juj 2 Mloc ./. Proof. Inequality (1.5.10) implies that there exists  2 Mloc./,   0 such that u D    C f . As .  /C  C, we may assume that (1.5.10) holds with equality. N @/. Put Let D be a smooth domain such that DN   and let 0 < < dist .D, u :D .J u/1D , f :D .J f /1D and  :D .J /1D . Then u D  C f in D so that, by (1.5.3),  .u /C  . C f /sign C.u /  . /C C f sign C.u /  .C / C f sign C .u /. Thus, for every non-negative  2 Cc1 .D/, Z Z Z .C / dx C f sign C .u /dx.  .u /C  dx  D

D

D

R

The first term on the right-hand side tends to D dC as ! 0. If jE0 j D 0 then Z Z .u /C  dx ! uC  dx D

and

Z

Z D

D

f sign C.u /dx !

D

f sign Cu.

If jE0 j ¤ 0 we apply the same argument to u  cn for a sequence ¹cn º decreasing to zero such that jEcn j D 0. This proves (1.5.11) which in turn implies that .uC / 2 Mloc./. Inequality (1.5.12) is proved in the same way, using (1.5.9).  Next we present an extension of Kato’s inequality to boundary value problems. For its proof we need a lemma.

24

Chapter 1 Linear second order elliptic equations with measure data

Lemma 1.5.8. Let u 2 L1 ./ and  2 M ./. Assume: (a) u   in . (b) There exists a C 2 uniform exhaustion ¹Dnº of  such that .with the notation n D @Dn/ Z jubn j dS < 1. (1.5.13) sup n

Then u possesses an M-boundary trace  2 M.@/ and there exists a non-negative  2 M ./ such that u D   .

(1.5.14)

Proof. Condition (a) implies that there exists a positive Radon measure  such that u D   . Let ¹Dn º be as in condition (b) and let Qn and Q be defined as in Lemma 1.3.5 when h D 1. By Corollary 1.2.3, with D D Dn and D Qn , (1.5.14) implies Z Z Z uQn dx D Qn d.  /  ubn dS. (1.5.15)  Dn

Dn

n

By (1.5.13) the second integral on the right-hand side is bounded uniformly with respect to n. Since u 2 L1./ it follows that the integral on the left-hand side of (1.5.15) is uniformly bounded. Therefore, as   0 and  2 M ./), Z Qn d  < 1. sup Dn

Since Qn ! Q   it follows that  2 M ./. Taking a subsequence if necessary, we may assume that ¹ubn HN 1 º converges weakly to a measure  2 M.@/: Z Z ubn h dS ! h d (1.5.16) n

@

N for every h 2 C./. N let ¹'nº and ' be as in Lemma 1.3.5 relative to ¹Dnº. Applying Given h 2 C 2./ once again Corollary 1.2.3 we obtain Z Z Z u'n dx D 'n d.  /  hubn dS. (1.5.17)  Dn

Dn

n

25

Section 1.5 Subsolutions, supersolutions and Kato’s inequality

Since u 2 L1./ and    2 M ./: Z Z  u'n dx ! u' dx, Dn

Z



Dn

Z 'n d.  / !

Hence, (1.5.16) and (1.5.17) imply Z Z Z u' dx D ' d.  /   





' d.  /.

hd,

(1.5.18)

@

and h D @n ' on @. This implies that u possesses an M-boundary trace given by  and completes the proof of the lemma.  Proposition 1.5.9. Let f 2 L1.; / and  2 M ./. Suppose that u 2 L1./ possesses an M-boundary trace  2 M.@/ and satisfies u  f C 

in .

(1.5.19)

Then, Z Z Z Z  uC  dx  f .sign C u/ dx C

dC  





If (1.5.19) holds with equality then Z Z Z Z juj dx  f .sign u/ dx C

d jj   





@

@

@n dC .

@n d jj,

(1.5.20)

(1.5.21)

N for every non-negative 2 C02./. Proof. By Corollary 1.5.7, (1.5.19) implies (1.5.11) and (1.5.12). Consequently there exist non-negative Radon measures ,   in  such that:

and

uC D f sign C u C C  

in 

juj D f sign u C C   

in .

(1.5.22)

(1.5.23) N Since tr@ u D , for every uniformly C exhaustion ¹Dnº of  and every h 2 C./, Z Z uhdS D h d. (1.5.24) lim 2

@Dn

@

Z

It follows that sup

@Dn

jujdS < 1.

By Lemma 1.5.8, uC possesses an M-boundary trace, say , Q and  2 M ./. Therefore, as uC satisfies (1.5.22), applying Proposition 1.3.7 to uC we obtain Z Z Z Z uC ' dx D 'd.C  / C ' f sign Cu dx  @n' d Q (1.5.25)  

for every ' 2



N C02./.



@

26

Chapter 1 Linear second order elliptic equations with measure data

As both u and uC possess M-boundary traces it follows that u too possesses an M-boundary trace, say   and we have: tr@ u D Q    D . Since both Q and   are non-negative it follows that C  . Q

(1.5.26)

As u possesses a boundary trace and satisfies (1.5.19), Lemma 1.5.8 implies that there exists a non-negative measure  2 M ./ such that u D f C   . Z

Put w1 :D

Z

Z @

v :D 

P .x, y/dC .y/,

G.x, y/f .y/dy C

Z

w2 :D



@

P .x, y/d .y/,

G.x, y/d.  /.

Then u D w1  w2 C v. Since both w1 and w2 are non-negative it follows that uC  w1 C vC,

u  w2 C v . Z

Z

As jv.x/j 



G.x, y/jf .y/jdy C



G.x, y/d j  j,

we have tr@ jvj D 0. Therefore tr@ uC  tr@ w1 , i.e., C  . Q

(1.5.27)

Inequalities (1.5.26) and (1.5.27) imply Q D C .

(1.5.28)

This fact and (1.5.25) imply inequality (1.5.20). If, in addition, u D f C  we apply (1.5.20) to u and obtain, Z Z Z Z  u  dx   f .sign  u/ dx C

d C @n d 





(1.5.29)

@

N Adding up inequalities (1.5.20) and (1.5.29) we for every non-negative 2 C02 ./. obtain (1.5.21).  Corollary 1.5.10. If u is a supersolution and v a subsolution of (1.2.4) then u  v.

Section 1.5 Subsolutions, supersolutions and Kato’s inequality

27

Proof. v  u is a subsolution of the problem w D 0,

tr@ w D 0.

Therefore .v  u/C is a subsolution of this problem, i.e., Z  .v  u/C  dx  0 

for every 2

Then

R

N C02 ./

such that  0. Let be the solution of the problem  D 1 in ,

D 0 on @.

 .v

 u/C dx  0 which implies .v  u/C 0.



Corollary 1.5.11. Suppose that u 2 L1loc ./ satisfies u   where  2 Mloc ./. Let D b  be a domain of class C 2 and denote by wD the L1 weak solution of the problem w D  in D, w D u on @D. Then, u  wD . Proof. There exists a non-negative Radon measure  such that u D   . By Corollary 1.2.3 

Z

Z D

u dx D

D

Z

d.  / 

@D

u@n dS,

N Now the inequality u  wD follows from Corolfor every non-negative 2 C02 .D/. lary 1.5.10.  Corollary 1.5.12. Suppose that u 2 L1./ satisfies u   for some  2 M ./. In addition suppose that there exists U 2 L1./ such that U D ,

u  U.

Then there exists a solution w 2 L1 ./ of w D  which is minimal among all solutions majorizing u. Proof. Let ¹Dnº be a uniformly C 2 exhaustion of . If wDn is defined as in the previous corollary then ¹wDn º is increasing and majorized by U . Therefore w D lim wDn is the minimal solution above u.



28

Chapter 1 Linear second order elliptic equations with measure data

1.6 Boundary Harnack principle In this section we describe a version of the boundary Harnack principle for operators with ‘nice’ coefficients. A detailed proof of the boundary Harnack principle for harmonic functions in bounded Lipschitz domains can be found in [5]. The following more general version, valid in Lipschitz domains, was proved in [2, Theorem 5.1]. Let L denote an uniformly elliptic operator in  of the form, X X ai ,j @2=@xi @xj C bi @=@xi C h (1.6.1) LD i ,j

i

N for some ˛ 2 .0, 1 (i.e Holder continuous with exponent where ai ,j , bi , h 2 C ./ ˛). Let  denote the ellipticity constant and let M be a bound for the norms of the N For P 2 @, we denote by nP the unit normal to @ at P , coefficients in C 0,˛ ./. pointing into the domain. In the case of a bounded Lipschitz domain , the normal may not be defined at some points. Instead we introduce the notion of an approximate unit normal defined below. Given A  S N 1 and r0 > 0 put CA :D ¹x 2 RN : x=jxj 2 Aº, CA,r0 D CA \ Œjxj  r0 . N 1 :D As  is a bounded Lipschitz domain, there exists an open spherical cap A  SC N 1 \ Œx1 D 0 and r0 > 0 such that, for every P 2 @, a rigid map TP mapping S the origin to P maps C A,r0 to a truncated cone contained in  [ ¹P º. Let y be the 0,˛

!

0 D PP 0 is called an center of the cap A and put P 0 D TP .y/. Then the unit vector nP approximate unit normal at P . For P 2 @, let T .P , r , r 0/ denote the truncated spherical cylinder of radius r 0 0 whose axis is the segment ŒP  r 0 nP , P C r 0 nP . For every P 2 @ let  P be a Euclidean set of coordinates centered at P such that the 1P axis points in the direction 0 P . Put Q P D .2P , : : : , N /. There exist r1 > 0 and r2 > 0 such that, for every of nP P 2 @,  \ T .P , r1 , r2/ D ¹ P : jQ P j < r1 , f P .Q P / < 1P < r2º where f P is a bounded uniformly Lipschitz function in RN such that f P .P / D 0 and the Lipschitz norm is bounded by a constant independent of P .

Theorem 1.6.1. Let P 2 @ and let T .P , r , r 0 / be a cylinder as described above such that r < r1, r 0 < r2 and r C r 0 is bounded by a constant which depends only on , ˛, M . Then there exists a constant C depending only on , ˛, M and r=r 0 such N of positive solutions of Lw D 0 vanishing that, for every pair u, v 2 C 2./ \ C./ on @ \ T .P , r , r 0 / (1.6.2) u.x/=v.x/  C u.P 0 /=v.P 0 /, 8x 2 T .P , r=2, r 0=2/. r0 0 0 Here P D P C 2 nP .

29

Section 1.6 Boundary Harnack principle

In the case of domains with C 2 boundary (which is the basic assumption in the present text) the boundary Harnack principle can be derived using the Hopf principle. We provide a proof in this case. Proposition 1.6.2. Let f 2 C 2 .Rn1 / be a function vanishing at the origin and put fr .x/ D r1 f .r x/. Denote Mfr :D kfr kC 2 .BN N 1.0// 2

and let r be the modulus of continuity of D 2 fr in BN 4N 1 .0/. Denote Q 2 R RN 1 : jj Q < r , j1 j < r º T .0, r / D ¹ D .1 , /

(1.6.3)

where 2 .0, 2/ is a number such that  .2 supBN N 1.0/ jrf j/1 . Thus, 4

sup jf .Q /j < r=2 8r 2 .0, 2.

Q j j<r

Let r 2 .0, 1 and put Q D D ¹ 2 T .0, 2r / : 1 > f ./º. Let L be a uniformly elliptic operator in DN of the form X X LD ai ,j @2 =@xi @xj C bi @=@xi C h, i ,j

i

N and h  0. Let M r be a with ellipticity constant , such that ai ,j , bi , h 2 C 0,˛ .D/ 2 0,˛ N bound of the norms of ai ,j , r bi , r h in C .D/. Assume: u 2 W 2,p .D/ for some p > N , u > 0 in D, u is normalized by u.r=2, 0, : : : , 0/ D 1 and u satisfies Lu D 0 in D,

Q jj Q  r º. u D 0 on @1D :D ¹ : 1 D f ./,

(1.6.4)

Then, for r  1, 1 C dist ., @1 D/  u./  dist ., @1 D/ 8 2 D \ T .0, r / rC r and C is a constant depending only on , ˛, M r , Mfr and r . Proof. If vr is the function given by vr .x/ D

1 u.r x/ r2

(1.6.5)

30 and

Chapter 1 Linear second order elliptic equations with measure data

Lr :D 

X i ,j

air,j @2 =@xi @xj C

X

r bir @=@xi C r 2hr

(1.6.6)

i

air,j .x/ D ai ,j .r x/, bir .x/ D bi .r x/, hr .x/ D h.r x/ 8x 2

1 D r

then

1 1 on @1.D/. Lr vr D 0 in D, u D 0 r r Furthermore, for r  1, Mr is a bound of the norms of air,j , r bir , r 2hr in C 0,˛ . r1 D/ and Mfr  Mf1 , r  1 . Therefore it is sufficient to prove the proposition in the case r D 1. By negation, suppose that (1.6.5) (with r D 1) is not valid. Then there exists a sequence of domains ¹Dn º of the form Q Dn D ¹ 2 T .0, 2/ : 1 > fn ./º,

such that ¹fn º is bounded in C 2 .BN 2N 1 .0// and ¹D 2fn º is uniformly continuous in this ball, and corresponding sequences of operators ¹Lnº with uniformly bounded ellipticity constants and solutions ¹unº satisfying the assumptions of the proposition with uniform bounds M and Mf such that ´ 1 if right-hand side of (1.6.5) fails un .n / ! (1.6.7) dist .n , @1 Dn / 0 if left-hand side of (1.6.5) fails for some sequence ¹n º  Dn \ T .0, 1/. We extend un to a function uQ n 2 W 2,p .T .0, 2// such that kuQ n kW 2,p .T .0,2//  c kuQ n kW 2,p .Dn / . Extracting a subsequence we may assume that fn ! f in C 2.BN 2N 1 .0//, uQ n ! uQ in W 2,p1 .T .0, 2// for some p1 > N and Ln ! L such that (1.6.4), with u D ub Q D holds in Q D D ¹ 2 T .0, 2/ : 1 > f ./º and L, u, f satisfy all the assumptions of the proposition. Furthermore we may assume that n !  2 D \ TN .0, 1/. As u.1=2, 0, : : : , 0/ D 1, u is a positive solution. Therefore, as un ! u uniformly in every compact subset of D, (1.6.7) implies that  2 @1D. Furthermore, it follows from (1.6.7) that there exists a sequence ¹nº such n n that n 2 Dn \ T .0, 1/, n !  and either @u .n / ! 1 or @u .n / ! 0. But @ 1 @ 1 2,p1 1 N .T .0, 2/ implies un ! u in C .T .0, 2//. Therefore convergence to uQ n ! uQ in W @u infinity is impossible. On the other hand, convergence to zero implies that @ ./ D 0, 1 which contradicts the strong maximum principle (Hopf’s lemma). 

31

Section 1.6 Boundary Harnack principle

Corollary 1.6.3. Let  be a domain of class C 2 and E a compact subset of @. Let u1 , u2 be positive functions in C 2 ./ such that Lm .um / D 0 in , where Lm D 

X

um D 0 on @ n E,

ai ,j ,m @2 =@xi @xj C

X

i ,j

(1.6.8)

bi ,m @=@xi C hm ,

i

N n E/ and hm  0. with ellipticity constant , such that the coefficients are in C 0,˛ . For y 2 @, let  D  y and Fy be as in Definition 1.1.1. Denote Q 2 R RN 1 : jj Q < r , j1 j < r º T .0, r / D ¹ D .1 , / and T .y, r / :D y C T .0, r /. Here 2 .0, 2/ is a number such that

 .2

sup BN 4N 1 .0/

jrFy j/1

8y 2 @.

Let 0 < r0 be as in (1.1.2) and denote r .y/ D

1 min.dist .E, y/, r0 / 8

8y 2 @.

For y 2 @ n E, let My be the maximum of the norms of ai ,j ,m , r .y/bi ,m, r .y/2 hm in C 0,˛ . \ B4r .y/.y//. Then 1 .x/ um .x/ .x/  C C r .y/ um .y  r .y/ny / r .y/

8x 2  \ T .y, r .y//, m D 1, 2 (1.6.9)

where ny is the unit normal on @ at y pointing outwards and c D c.y/ is a constant depending on My , on the C 2 characteristic of the domain and on the modulus of continuity associated with the C 2 representation of @ \ B4ry .y/. Consequently, 0 u1 .x/ 1 u1 .x 0 / 2 u1 .x /   c c 2 u2 .x 0 / u2 .x/ u2 .x 0 /

8x, x 0 2  \ T .y, r .y//

(1.6.10)

Proof. Applying (1.6.5) to each of the normalized solutions vm :D

um , um .y  r=2ny /

m D 1, 2

in T .y, r .y// we obtain (1.6.9) which in turn implies (1.6.10).



32

Chapter 1 Linear second order elliptic equations with measure data

1.7 Notes The three classic results presented in this section – the Herglotz–Doob theorem, Kato’s inequality and the boundary Harnack principle – have been key factors in the development of modern PDE theory. Harnack inequalities originated in a paper from 1887 by C. G. Harnack dealing with positive harmonic functions in a disk. J. Serrin [104] extended the inequality to elliptic linear equations (in non-divergence form) with continuous coefficients. Later the inequality was extended by J. Moser [91] to equations in divergence form with bounded coefficients. The Boundary Harnack Principle (or BHP) deals with estimates of the ratio of two positive solutions vanishing on a portion of the boundary. The first version of BHP, for harmonic functions in Lipschitz domains, is due to L. Carleson [29]. A detailed proof of BHP for harmonic functions in bounded Lipschitz domains can be found in [5]. Extensions of this important principle to more general elliptic equations were obtained by several authors. We mention in particular the results of A. Ancona [2] and P. Bauman [13]. The version stated in Section 1.6 is proved in [2]. The characterization of a family of solutions by means of a measure boundary trace was introduced by G. Herglotz [56] with respect to positive harmonic functions and extended by J.L. Doob to positive superharmonic functions. The definition of M-boundary trace for arbitrary functions in Wloc1,p given in Definition 1.3.6 was introduced in [73]. There it was used in order to obtain the related result on boundary value problems, Proposition 1.3.7. This proposition is a very useful tool in treating boundary value problems, as will be seen in the following two chapters. The proposition was extended to boundary value problems in Lipschitz domains in [86]. Kato’s inequality is used in [59] in order to prove the self adjointness of operators of the form 2 m  X @  ibj C q.x/ T D @xj j D1

for suitable choice of potentials q. The extension of Kato’s inequality, Proposition 1.5.9, is due to H. Brezis [21], but the proof presented here is different. We do not deal here with the important subject of harmonic functions in non-smooth domains. The main historical steps in this subject are due to H. Poincaré who initiated the “balayage” method [98], O. Perron who is at the origin of the construction of the solution of the Dirichlet problem as a supremum of subharmonic minorants [97] and N. Wiener who found the necessary and sufficient conditions on the regularity of the domain, expressed in terms of an infinite sum involving the Newtonian capacity C1,2 [125].

Chapter 2

Nonlinear second order elliptic equations with measure data 2.1 Semilinear problems with L1 data We study the semilinear boundary value problem, u C g ı u D  in ,

u D  on @

(2.1.1)

where  is a bounded C 2 domain in RN , g is defined on  R and g ı u.x/ D g.x, u.x//. We assume that g satisfies the following conditions: .a/

g.x, / 2 C. R/, g.x, 0/ D 0, g.x, / is non-decreasing,

.b/

g., t / 2 L1 .; / 8t 2 R.

(2.1.2)

The family of functions satisfying these conditions will be denoted by G0 D G0 ./. Observe that if g 2 G0 then the function g  given by g  .x, t / D g.x, t /,

(2.1.3)

is also in G0. Definition 2.1.1. Assume that  2 M ./ and  2 M.@/. A function u 2 L1./ is a weak solution of problem (2.1.1) if g ı u 2 L1 .; / and Z Z Z .u C .g ı u/ / dx D

d  @n d, (2.1.4) 



@

N C02 ./. 1

for every 2 A function u 2 L ./ is a weak subsolution of the problem if g ı u 2 L1 .; / and Z Z Z .u C .g ı u/ / dx 

d  @n d, (2.1.5)  2 N C0 ./ such that



@

for every 2  0. A weak supersolution of the problem is defined in the same way with the inequality inverted. A function u 2 L1loc./ is a weak supersolution .resp. subsolution / of the equation in (2.1.1) if g ı u 2 L1loc./ and u C g ı u   .resp. u C g ı u  / in the sense of distributions. We shall denote by 0 the solution of the problem  D 1 in ,

D 0 on @.

(2.1.6)

34

Chapter 2 Nonlinear second order elliptic equations with measure data

It will be often used as a test function, or in the construction of more complex test N  > 0 in  and @n  > c > 0 on @. functions. Note that 0 2 C02./, 0 0 Therefore c   0  2c 2 in a neighborhood of the boundary. Where there is no danger of confusion, we shall drop the superscript . The first assertion in the following proposition is essentially due to Brezis and Strauss [26] and the others to Brezis [21]. Proposition 2.1.2. The boundary value problem u C g ı u D f uD

in , on @

(2.1.7)

with g 2 G0, f 2 L1 .; / and  2 L1.@/ possesses a unique weak solution u 2 L1 ./ . There exists a constant C depending only on  such that,   (2.1.8) kukL1 ./ C kg ı ukL1 .;/  C kf kL1.;/ C kkL1.@/ . If ui are weak solutions corresponding to data fi , i , i D 1, 2 then ku1  u2 kL1./ C kg ı u1  g ı u2 kL1 .;/    C kf1  f2 kL1.;/ C k1  2 kL1 .@/ . If, in addition, f1  f2 and 1  2 then u1  u2 . If u is a weak solution of (2.1.7) then Z Z Z .uC  C .g ı u/C / dx  f .sign Cu/ dx  



Z

Z 

.juj C jg ı uj / dx 



@

(2.1.9)

C@n dS, (2.1.10)

Z f .sign u/ dx 

@

jj@n dS,

(2.1.11)

N such that  0. for every 2 C02 ./ Proof. The uniqueness, monotonicity and inequalities (2.1.8)–(2.1.11) are a consequence of Proposition 1.5.9. We observe that inequality (2.1.9) is obtained by applying (1.5.21), with 0 D 0 as test function, to the problem .u1 u2 / D f  :D f1f2 gıu1 Cgıu2

in ,

u1 u2 D 12

on @.

The monotonicity follows by applying (1.5.20), with 0 as test function, to this problem. Clearly, (2.1.9) implies (2.1.8) and uniqueness. In addition (2.1.10) and (2.1.11) follow from (1.5.20) and (1.5.21) respectively.

Section 2.1 Semilinear problems with L1 data

35

We turn to the proof of existence starting with the case f 2 L1 ./ and  D 0. To simplify the presentation we assume that g D g  , i.e., jg.x, t /j D g.x, jt j/. The modifications needed for the general case are straightforward. Put gn :D max.jgj, n/sign g and let Gn .x, / be the primitive of gn .x, / such that Gn .x, 0/ D 0. Consider the functional Z  Z  1 In .u/ :D jruj2 C Gn ı u dx  uf dx (2.1.12)  2  over the space W01,2./. Since Gn is a non-negative function, the functional is coercive. By Fatou’s lemma, if vm * u weakly in W01,2./ then Z Z Gn ı u dx  lim inf Gn ı vm dx. 

m!1



Therefore In is weakly lower semi-continuous and the variational problem min In .u/ possesses a solution un 2 W01,2 ./. The minimizer un is also a weak solution of the boundary value problem u C gn ı u D f uD0

in , on @.

(2.1.13)

Consequently, by (2.1.8), the sequences ¹un º and ¹gn ı un º are bounded in L1 ./ and L1.; / respectively. Assume for a moment that f  0. Then In .jun j/  In .un / so that un  0. In addition ¹un º is monotone decreasing so that un ! u in L1./. This fact also implies that gn ıun ! gıu. Indeed, if uk .x0/ D: ak then, for m > max.k, ak /, gm ıum .x0 / D g ı um .x0/ ! g ı u.x0 /. Further, the sequence is dominated by g ı V where V is the solution of v D f in  with zero boundary data. (Recall that we still assume f 2 L1 ./.) Thus u is the weak solution of the boundary value problem u C g ı u D f uD0

in , on @.

(2.1.14)

Dropping the assumption that f  0, observe that jun j  uQ n where uQ n is the solution of (2.1.13) with f replaced by jf j. Furthermore ¹un º is bounded in W 1,p or even in C 1. Hence un ! u in L1./ and (taking a subsequence if necessary) un ! u a.e. Hence ¹gn ı un º converges a.e. and is dominated by ¹gn ı uQ n º. Therefore u is a weak solution of (2.1.14). This was proved for f 2 L1 ./. By approximation, using (2.1.9), we obtain existence for every f 2 L1 .; /. If  is not necessarily zero but  2 C 2.@/ the problem can be reduced to the previous special case. Let v be the harmonic function in  with boundary value . Then (2.1.7) can be rewritten as w C gQ ı w D fQ wD0

in , on @

36

Chapter 2 Nonlinear second order elliptic equations with measure data

with g.x, Q t / D g.x, t Cv.x//g.x, v.x// and fQ D f gıv. Note that fQ 2 L1 .; / and that gQ satisfies conditions (2.1.2). Therefore (2.1.7) possesses a weak solution whenever f 2 L1.; / and  2 C 2 .@/. In order to prove existence in the general case, we approximate  in L1.@/, by a sequence ¹n º in C 1 .@/. If un is the solution corresponding to data f , n then, by (2.1.9), un ! u in L1./, g ı un ! g ı u in L1.; / and u is a weak solution of (2.1.7).  Proposition 2.1.3. Suppose that g 2 G0 . Let u, v 2 L1 ./ and suppose that u is a weak subsolution and v a weak supersolution of problem (2.1.4) with ,  replaced by i , i , i D 1, 2 respectively. If 1  2 and 1  2 then u  v. Proof. Put w D .u  v/. The assumption implies that Z Z Z .w C .g ı u  g ı v/ / dx 

d.1  2 /  



@

@n d.1  2 /  0

N such that  0. By Proposition 1.5.9 for every 2 C02 ./ Z .wC  C .g.x, u/  g.x, v//.sign C w/ / dx  0, 

for every as above. Since g.x, / is non-decreasing, .g.x, u/  g.x, v//.sign C w/  0. Choosing 0 as a test function we obtain wC D 0.



2.2 Semilinear problems with bounded measure data In contrast to the semilinear problem with L1 data, the problem with measure data does not necessarily possess a solution. However, if a solution exists then it is unique and inequality (2.1.9) remains valid. Proposition 2.2.1. Consider the boundary value problem u C g.x, u/ D 

in ,

uD

on @

(2.2.1)

with g 2 G0 ,  2 M ./ and  2 M.@/ The problem possesses at most one weak solution u 2 L1./. If a solution exists then   kukL1 ./ C kg ı ukL1 .;/  C kkM ./ C kkM.@/ (2.2.2) where the constant C depends only on g and .

37

Section 2.2 Semilinear problems with bounded measure data

If ui are weak solutions corresponding to data i , i , i D 1, 2 then ku1  u2 kL1./ C kg ı u1  g ı u2 kL1 .;/    C k1  2 kM ./ C k1  2 kM.@/ . (2.2.3) If, in addition, 1  2 and 1  2 then u1  u2 . If u is a weak solution of (2.2.1) then, Z Z Z .uC  C .g ı u/C / dx 

dC  

Z

Z



.juj C jg ı uj / dx 





Z

d jj 

@

@

@n dC ,

@n d jj,

(2.2.4) (2.2.5)

N such that  0. for every 2 C02 ./ The proof runs along the same lines as that of the corresponding statements in Proposition 2.1.2. Of course, in contrast to Proposition 2.1.2, here there is no claim of existence. Definition 2.2.2. Given g 2 G0 we denote by Mg ./ the set of measures  2 M ./ such that the boundary value problem u C g ı u D  tr@ u D 0

in 

(2.2.6)

possesses an L1 weak solution. If  2 Mg ./ we say that  is a g-good measure in . We denote by Mg .@/ the set of measures  2 M.@/ such that the boundary value problem u C g ı u D 0 in  (2.2.7) tr@ u D  possesses an L1 weak solution. If  2 Mg .@/ we say that  is a g-good measure on @. Finally, the set of pairs of measures ., / 2 M ./ M.@/ such that (2.2.1) N if ., / 2 M ./ M.@/ we say possesses a solution will be denoted by Mg ./; N that it is a g-good pair in . When there is no danger of confusion we shall simply say ‘a good measure’ instead of ‘a g-good measure’. N Denote Proposition 2.2.3. Let g 2 G0./. Let ¹.i , i /º be a sequence in Mg ./. by ui the solution of (2.2.1) with  D i ,  D i .

38

Chapter 2 Nonlinear second order elliptic equations with measure data

(a) If ¹i º is bounded in M ./ and ¹i º is bounded in M.@/ then ¹ui º is bounded in L1./ and possesses a subsequence which converges in L1loc./ and pointwise a.e. to a function u 2 L1./. (b) Suppose that i *  and i *  weakly relative to Cc ./ and C.@/ respectively. In addition suppose that at least one of the following conditions holds: (i) i !  in M ./, i !  in M.@/ (convergence in norm). N such that ji j  N and ji j  . N (ii) There exists ., N N / 2 Mg ./ (iii) ¹i º and ¹i º are non-decreasing sequences. Then ui ! u in L1 ./ and pointwise a.e., g ı u 2 L1 .; / and u is the solution of (2.2.1). Proof. (a) By Proposition 2.2.1, ¹ui º is bounded in L1./ and ¹g ı ui º is bounded in L1.; /. Therefore, by Theorem 1.2.2, ¹ui º is bounded in Wloc1,p ./. This implies assertion (a). The weak convergence assumed in (b) and any of the conditions (i)–(iii) implies that every subsequence ¹uik º which, according to (a), converges pointwise a.e., actually converges in L1./ and also that ¹g ı uik º converges in L1.; /. Moreover, if u D lim uik , then u is a solution of (2.2.1). The uniqueness of the solution implies that ui ! u.  Theorem 2.2.4. Consider the problem u C g ı u D , uD

in , on @

(2.2.8)

for g 2 G0 ,  2 M.@/ and  2 M ./. Assume that there exist a weak subsolution v and a weak supersolution w of this problem such that v  w. Then (2.2.8) possesses an L1 weak solution. Remark 2.1. Recall that the assumption on v and w includes g ı w 2 L1.; /, g ı v 2 L1.; /,

v 2 L1./, w 2 L1 ./.

(2.2.9)

Proof. The space M./ is dense in M ./. Indeed, if ¹n º is an exhaustion of  then, for  2 M.; /, n :D 1n H) k  n kM.;/ ! 0. Therefore it is sufficient to prove the theorem in the case  2 M./. Accordingly, in the rest of the proof we assume that  2 M.

39

Section 2.2 Semilinear problems with bounded measure data

Let gQ be defined by

8 ˆ
if v.x/  t  w.x/, if v.x/  t , if t  w.x/.

(2.2.10)

We note that g.x, Q 0/ is not identically zero. But g.x, Q 0/ D 0 whenever v.x/  0  w.x/ and gQ possesses all the other properties required of functions in G0. Let V1, V2 be the weak solutions of the following problems: V1 D jj C jg ı wj C jg ı vj V2 D 0

in , in ,

V1 D 0 V2 D 

on @, on @.

For every  2 L1./ let z :D T  be the weak solution of z C gQ ı  D , zD

in , on @.

(2.2.11)

By (2.2.10) and the maximum principle,  2 L1./ H) jgQ ı j  jg ı wj C jg ı vj H) jT j  V :D V1 C V2. Therefore, T .L1.//V2 is a bounded set in W 1,p ./ for some p > 1. Consequently T .L1.// is a compact set in L1./. By Schauder’s fixed point theorem T has a fixed point u. Thus the problem u C gQ ı u D , uD

in , on @

(2.2.12)

possesses a weak solution. Up to this point we did not use the assumption that v and w are weak sub- and supersolutions of (2.2.8). We use this assumption now and observe that v and w are also weak sub- and supersolutions of the modified problem (2.2.12). Therefore, by Proposition 2.1.3, the solution u of this problem satisfies v  u  w and consequently u is a weak solution of (2.2.8) as well. Observe that in the proof of Proposition 2.1.3 we only made use of those properties of the nonlinearity which are satisfied by g. Q  Definition 2.2.5. Let g 2 G0 ./. A pair of measures ., / 2 M ./ M.@/ is called g-admissible if the solutions u.˙/ of the linear problems u D ˙ u D ˙ satisfy

g ı u.C/ 2 L1 .; /,

with g  as in (2.1.3).

in , on @, g  ı u./ 2 L1 .; /

(2.2.13)

40

Chapter 2 Nonlinear second order elliptic equations with measure data

If ., 0/ (resp. .0, /) is g-admissible we shall say that  (resp. ) is g-admissible. (Note that the function u :D u.C/  u./ satisfies uC :D max.u, 0/  u.C/ .) Remark 2.2. Let U denote the weak solution of the linear problem U D jj

U D jj

in ,

on @.

(2.2.14)

Then u.C/  U and u./  U . Therefore, if g ıU 2 L1.; / and g  ıU 2 L1 .; / then ., / is g-admissible. The following is a simple consequence of Theorem 2.2.4. Corollary 2.2.6. Let g 2 G0./ and assume that ., / 2 M ./ M.@/ is gadmissible. Then (2.2.1) is solvable. Proof. With the notation of Definition 2.2.5, u.C/ is a weak supersolution and u./ is a weak subsolution of problem (2.2.1). Obviously u./  u.C/ . Therefore the assertion of the theorem is an immediate consequence of Theorem 2.2.4.  Theorem 2.2.7. Consider the equation u C g ı u D 

(2.2.15)

where  2 Mloc./ and g 2 G0. Suppose that v 2 L1loc./ is a subsolution and w 2 L1loc./ is a supersolution of (2.2.15) and that v  w. Then wQ :D sup¹z 2 L1loc ./ : z  w, z C g ı z  º (2.2.16) is a solution of (2.2.15) and it is the largest solution dominated by w. Similarly vQ :D inf¹z 0 2 L1loc./ : v  z 0 , z 0 C g ı z 0  º

(2.2.17)

is a solution of (2.2.15) and it is the smallest solution that dominates v. Assume, in addition, that g ı w  g ı v 2 L1.; /. Then: (i) w  v, w  vQ and wQ  v possess M-boundary traces on @ and tr@ .w  v/ Q D tr@ .w  v/ D tr@ .wQ  v/.

(2.2.18)

(ii) tr@ v exists if and only if tr@ w exists. When this is the case then Q tr@ w D tr@ w,

tr@ v D tr@ v. Q

(2.2.19)

Finally, if gıw, gıv 2 L1 .; /,  2 M ./ and either v or w has an M-boundary trace then w, v, w, Q vQ have M-boundary traces and wQ C g ı wQ D  vQ C g ı vQ D 

in , in ,

wQ D tr@ w, vQ D tr@ v.

(2.2.20)

41

Section 2.2 Semilinear problems with bounded measure data

Remark 2.3. Recall that the statement ‘w is a supersolution of (2.2.15)’ includes the assumption that g ı w 2 L1loc./. Proof. For 0 < ˇ < ˇ0 consider the problem u C g ı u D  uDv

in Dˇ , on †ˇ :D @Dˇ .

(2.2.21)

Our assumptions on  and v, w imply that  2 M.Dˇ / and that v, w satisfy (2.2.9) in Dˇ and possess M-boundary traces on †ˇ . Furthermore, v and w (restricted to Dˇ ) are weak sub- and supersolutions of (2.2.21). Since v  w Theorem 2.2.4 implies that there exists a solution vˇ of this problem such that v  vˇ  w. If ˇ1 < ˇ2 then, by Corollary 1.5.11, in Dˇ2 w  vˇ 1  vˇ 2  v because vˇ1 b†ˇ2  vb†ˇ2 . Q Furthermore, if z 0 is a Therefore vQ :D limˇ !0 vˇ is a solution of (2.2.15) and v  v. 0 0 supersolution of the equation and v  z then vˇ  z in Dˇ ; hence vQ  z 0 . Thus vQ satisfies (2.2.17) and it is the smallest solution of (2.2.15) dominating v. The corresponding assertion for wQ is proved in the same way. Assume, in addition, that g ı w  g ı v 2 L1 .; /. Let V be the weak solution of u D g ı w  g ı v uD0

in , on @.

Then w  v C V is positive and superharmonic. Therefore, by Theorem 1.4.1, w  v C V 2 L1./ and possesses an M-boundary trace on @. Since tr@ V D 0, it follows that tr@ .w  v/ is defined. In the same way we conclude that wQ  v, w  vQ and wQ  vQ possess M-boundary traces on @. The construction of vQ and wQ implies (2.2.18). Finally (2.2.19) is an immediate consequence of (2.2.18). The last assertion of the theorem follows from the previous statements and Proposition 1.3.7.  Corollary 2.2.8. Suppose that w 2 L1loc ./ is a positive supersolution of equation u C g ı u D 0.

(2.2.22)

Q as defined in (2.2.16), is a solution of (2.2.22). Both w and wQ If g ı w 2 L1 then w, have M-boundary traces and tr@ w D tr@ w. Q Proof. This follows from the theorem with v D 0.



42

Chapter 2 Nonlinear second order elliptic equations with measure data

Theorem 2.2.9. Consider problem (2.2.8) for g 2 G0,  2 M.@/ and  2 M ./. If u is a weak subsolution of the problem then u possesses an M-boundary trace on @ and there exist non-negative measures  2 M ./ and  2 M.@/ such that u satisfies u C g ı u D    in , (2.2.23) uD  on @. Proof. By assumption, Z Z Z .u C .g ı u/ / dx 

d  



@

@n d,

(2.2.24)

N such that  0. This immediately implies that there exists a for every 2 C02 ./ positive Radon measure  such that u  g ı u D   

in 

(2.2.25)

in the distribution sense. Let ˇ 2 Cc1 ./ satisfy 0  ˇ  1 and ˇ D 1 on Dˇ . N such that  0 put Q ˇ D .1  ˇ /. Then (2.2.25) yields Given 2 C02./ Z Z .u. ˇ / C .g ı u/ ˇ / dx 

ˇ d.  / 



and (as u is a subsolution of problem (2.2.8)) Z Z Z Q Q Q

ˇ d  .u ˇ / C .g ı u/ ˇ / dx  



Adding the last two inequalities and observing that Z

Q ˇ d ! 0 as ˇ ! 0, @n Q ˇ D @n

@

@n Q ˇ d.

on @

ˇ

we obtain

Z

Z

I.u, / :D 

.u C .g ı u/ / dx 

Z 

ˇ d.  / 

@

@n d C ,

where ! 0 as ˇ ! 0. Since ˇ D 1 on Dˇ it follows that Z Z Z

d  I.u, / C

d  @n d C . Dˇ



@

The right-hand side is bounded by a constant independent of ˇ. Therefore Z

d < 1 

(2.2.26)

43

Section 2.3 Subcritical nonlinearities

N We conclude that for every non-negative 2 C02./.  2 M ./ and

Z

Z 

.u C .g ı u/ / dx 

Z

d.  / 



@

@n d,

(2.2.27)

for every as above. By (2.2.25), if 2 Cc1 ./ (not necessarily positive) then (2.2.27) holds with equality. Let w be the weak solution of the problem w D .  /  g ı u

in ,

wD

on @.

In view of (2.2.27), w  u and .w  u/  0. Thus w  u is a positive weak superharmonic function. By Theorem 1.4.1, w  u possesses a non-negative M-trace  2 M.@/. We conclude that u possesses an M-boundary trace given by tr@ u D   .

(2.2.28)

By Proposition 1.3.7, (2.2.25) and (2.2.28) imply that u is a weak solution of problem (2.2.23). 

2.3 Subcritical nonlinearities We start with some definitions and notations. Notation. (i)

For every  2 M.; / put

Z

GΠD

G., y/d.y/,

(2.3.1)



where G D G denotes Green’s function for  in . For every  2 M.@/ put

Z P ΠD

P ., y/d.y/,

(2.3.2)

@

where P D P denotes Poisson’s kernel for  in  . (ii) Let g 2 G0 . Denote by

g S : Mg ./ 7! L1./

the operator which assigns to a measure  2 Mg ./ the L1 weak solution of (2.2.6) and by g S@ : Mg .@/ 7! L1./ the operator which assigns to a measure  2 Mg .@/ the solution of (2.2.7).

44

Chapter 2 Nonlinear second order elliptic equations with measure data

(iii) If y 2 RN , ıy denotes the Dirac unit measure concentrated at y. For E  RN , D.E/ denotes the set of all (finite) linear combinations of Dirac measures supported in E. Definition 2.3.1. Let g 2 G0 ./. We say that g is subcritical relative to boundary data (resp. interior data) if Mg .@/ D M.@/ (resp. Mg ./ D M.; /). g We say that S is g-weakly continuous on M.; / if Mg ./ D M.; / and, for every tight sequence ¹m º  M.; / such that m *  weakly in M.; /, g g .m / ! g ı S ./ g ı S

in L1.; /.

(2.3.3)

g

We say that S@ is g-weakly continuous on M.@/ if Mg .@/ D M.@/ and, for every sequence ¹m º  M.@/ such that m *  weakly in M.@/, g

g

g ı S@ .m / ! g ı S@ ./

in L1.; /.

(2.3.4)

g is g-weakly continuous on M.; / then, for every bounded Remark 2.3.1. If S tight sequence ¹m º  M.; / g

g

m *  H) S .m / ! S ./ in L1./.

(2.3.5)

g .m /º is bounded in Wloc ./ for Indeed under these assumptions the sequence ¹S some p > 1 and consequently one can extract a subsequence that converges a.e. to a function v. By extracting a further subsequence one can assume that ¹jm jº converges weakly to a measure . By Theorem 1.2.5 1,p

GŒjm j ! GŒ Since

in L1 ./.

g

jS .m /j  GŒjm j g

it follows that ¹S .m /º is uniformly integrable in L1 ./. By Vitali’s theorem, g g ¹S.m /º ! v in L1./. Hence, by (2.3.3), v D S ./. Since the limit is independent of the subsequence, we obtain (2.3.5). g In the same way one verifies the corresponding statement for S@ : g is g-weakly continuous on M.@/ then, for every bounded sequence ¹m º  If S@ M.@/, g g (2.3.6) m *  H) S@ .m / ! S@ ./ in L1./. In this section we present sufficient conditions for subcriticality and weak continuity in the sense of the above definition. We begin by deriving some auxiliary results.

2.3.1 Weak Lp spaces p Denote by Lp w .; /,  2 MC ./, 1  p < 1, the weak L space defined as follows: a measurable function f in  belongs to this space if there exists a constant

45

Section 2.3 Subcritical nonlinearities

c such that f .a; / :D .¹x 2  : jf .x/j > aº/  cap

8a > 0.

(2.3.7)

The function f is called the distribution function of f (relative to ). For p  1 denote p Lp w .; / D ¹f Borel measurable : sup a f .a; / < 1º a>0

and

kf kLpw ./ D .sup ap f .a; //1=p .

(2.3.8)

a>0

The kkLp ./ is not a norm but, for p > 1, it is equivalent to the norm: w

²R

kf kLpw .;/ D sup

!

jf jd 

.!/1=p

0

³ : !  , ! measurable, 0 < .!/ < 1 . (2.3.9)

In fact the following inequality holds: kf kLp .;/  kf kLpw .;/  w

p kf kLp .;/ . w p1

(2.3.10)

(For details see [32, p. 300–311].) When  is the Lebesgue measure, we drop the reference to  in Lp w and f . Lemma 2.3.2. For 0  ˛ < N , let ˛ denote the function given by ˛ .x/ D jxjN C˛

8x 2 RN .

Then, for every y 2 RN and ˛ 2 Œ0, N /, N=.N ˛/ ˛ .  y/ 2 Lw ./.

If ˛ 2 .0, N /, there exists a constant c (independent of y) such that, k˛ .  y/kLN=.N ˛/./  c. w

(2.3.11)

Further, if ˛ 2 .0, N /, ˇ > 1 and ˛ C ˇ > 0 then, for every y 2 @, N Cˇ

˛ .  y/ 2 LwN ˛ .;  ˇ / and there exists a constant c (independent of y) such that, k˛ .  y/k

N Cˇ N ˛ .,ˇ / Lw

 c.

The constant c depends only on N , ˛, ˇ and diam .

(2.3.12)

46

Chapter 2 Nonlinear second order elliptic equations with measure data

Proof. One verifies in a straightforward manner that, for every ˛ 2 Œ0, N / k˛ .  y/kLN=.N ˛/.B

1 .y//

w

 1.

If ˛ > 0, this inequality and (2.3.10) imply (2.3.11). Next we verify (2.3.12). Given y 2 @ put Z fy .x/ D ˛ .x  y/ 8x 2 , y ./ D

.x/ˇ dx

Œfy > 

Then

0 :D  N ˛ 1

¹x 2  : fy .x/ > º  B 0 .y/, and, since y 2 @,

8 > 0.

jx  yj  .x/. Z

Therefore

y ./ 

N Cˇ

jx  yjˇ dx D c N ˛ .

B 0 .y/

(2.3.13)

If ! is a measurable subset of  such that j!j ¤ 0 and a positive number then Z Z Z ˇ ˇ fy .x/ dx  .x/ dx C fy .x/.x/ˇ dx. (2.3.14) !

!

Œfy >

In view of the assumption ˛ C ˇ > 0, (2.3.13) implies that lim R y .R/ D 0.

R!1

Therefore integration by parts yields Z Z ˇ fy .x/.x/ dx D  Œfy >

1 

Z  d y ./ D y . / C

1 

y ./d.

Therefore, combining (2.3.13), (2.3.14), using once more the fact that ˛ C ˇ > 0, we obtain Z Z 1 Z fy .x/.x/ˇ dx  .x/ˇ dx C y . / C

y ./d ! !  Z Z 1 N Cˇ N Cˇ  . .x/ˇ dx C c  N ˛ / C c  N ˛ d  ! Z N Cˇ ˇC˛  . .x/ˇ dx C c  N ˛ / C c 0  N ˛ . !

Choosing

Z

ˇ

:D

 N ˛

N Cˇ

 dx !

we obtain

Z !

fy .x/.x/ˇ dx  C

Z

 ˇC˛ .x/ˇ dx

N Cˇ

!

where C depends only on N , ˇ and ˛. In view of (2.3.9) this is equivalent to (2.3.12).

47

Section 2.3 Subcritical nonlinearities

2.3.2 Continuity of G and P relative to Lpw norm Lemma 2.3.3. The following mappings are continuous relative to the norm topologies: N Cˇ

G : M./ 7! LwN 2 .;  ˇ /,

.i/

N Cˇ N 1

G : M ./ 7! Lw

.ii/ and

(2.3.15)

ˇ

.;  / 8ˇ > 1

N Cˇ

P : M.@/ 7! LwN 1 .;  ˇ /

(2.3.16)

8ˇ > 1.

The norms of these mappings depend only on N , ˇ, diam  and the constants c1 ./ D sup jx  yjN 1 G.x, y/= min..x/, .y// c2 ./ D sup jx  yjN 2 G.x, y/ 8.x, y/ 2  , x ¤ y in (2.3.15), c10 ./ D sup jx  yjN P .x, y/=.x/, c20 ./ D sup jx  yjN 1 P .x, y/

8.x, y/ 2  @.

in (2.3.16). By (1.1.6) and (1.1.7) these are finite. Proof. Let  2 M./. To simplify notation we assume that  is positive. Otherwise we replace  by jj. We consider  as a measure in RN vanishing outside . Assume 0 < ˛ < N , 1 < ˇ, ˛ C ˇ > 0. For every measurable set !   put .!/ D

Z  ˇ dx. !

Assuming j!j ¤ 0, Z ˇC˛ N Cˇ .!/ .˛  /.x/ˇ dx !

Z   ˇC˛ N Cˇ  sup .!/ ˛ .x  y/.x/ˇ dx kkM./ . !

y2

Hence, by (2.3.12), Z ˇC˛ .!/ N Cˇ .˛ /.x/ˇ dx  sup k˛ .  y/k !

y2

N Cˇ N ˛ .;ˇ / Lw

./  c0 kkM./

which is equivalent to k.˛  /k

N Cˇ N ˛ .;ˇ / Lw

Here c0 depends only on N , ˛ and diam .

 c0 kkM./ .

(2.3.17)

48

Chapter 2 Nonlinear second order elliptic equations with measure data

The same argument shows that, if  is a measure in M.@/ – which we consider as a measure in RN supported in @ – then k˛  k

 c0 kk M.@/

N Cˇ N ˛ .;ˇ / Lw

8 2 M.@/.

(2.3.18)

By (1.1.6), G.x, y/  c2 ./2 .x  y/,

G.x, y/  c1 ./.y/1 .x  y/.

Consequently, GŒjj  c2 ./2  jj

8 2 M./

and GŒjj  c1 ./1  .jj/ 8 2 M./. Therefore, by (2.3.17), applied to 2  , kGŒk

N Cˇ N 2 ./ Lw

 c0 c2 ./ kkM./

8 2 M./.

(2.3.19)

8 2 M.; /.

(2.3.20)

By (2.3.17), applied to 1  ./ kGŒk

N Cˇ N 1 .;ˇ / Lw

 c0 c1 ./ kkM.;/

This proves (2.3.15) Similarly, if  2 M.@/ then, by (1.1.7), P Œjj  c20 ./1  jj. Here as before,  is considered as a measure in M.RN / supported in @. Therefore, by (2.3.18), kP Œk

N Cˇ N 1 .;ˇ / Lw

 c0 c20 ./ kk M.@/

8 2 M.@/.

(2.3.21) 

This proves (2.3.16).

2.3.3 Continuity of a superposition operator Theorem 2.3.4. Let h 2 C.R/ be an odd monotone increasing function and let  be a positive measure in M./. Assume that, for some p 2 .1, 1/, Z 1 h.r 1=p /dr < 1. (2.3.22) 0 p If ¹wm º is bounded in Lw .; / then ¹hıwm º is uniformly integrable in L1.; /. The

modulus of uniform integrability depends on h, N , p and the bound of the sequence p in Lw .; / but not on the sequence itself, nor on  and . Consequently, if ¹wm º is bounded in Lp w .; / and wm ! w -a.e. then h ı wm ! 1 h ı w in L .; /.

49

Section 2.3 Subcritical nonlinearities

Proof. For any real measurable function f in  denote E t D E t .f / :D ¹x 2  : jf .x/j > t º and .t / D .E t / 8t  0. We extend f to the entire space RN by setting f D 0 outside . A classical result states (see [111], diff.; Ch.1): Z 1 Z h.jf .x/j/d  D  h.t /d.t / 8a  0. (2.3.23) a

Ea

We start with the following: p Assertion. If f 2 Lw .; /,

Z lim

t !1 E t

h.jf .x/j/d  D 0.

(2.3.24)

The rate of convergence depends on h, N , p and kukLpw .;/, but not specifically on u,  or . Since h is monotone, integration by parts yields: Z 1 Z 1 (2.3.25) h.t /d.t /  h.a/.a/ C .t /dh.t /.  a

Assuming that f 2

a

Lp w .; /

we have, p

.t /  c.p/ kf kLp .;/ t p :D A.f /t p w

Therefore, Z Z M .t /dh.t /  A.f / a

M

t p dh.t /

a

D A.f /.M

p

h.M /  a

p

Z h.a// C pA.f /

M

h.t /t p1 dt .

a

Assumption (2.3.22) implies that lim inf .M /h.M /  A.f / lim inf M p h.M / D A.f / lim inf r h.r 1=p / D 0. M !1

r !0

M !1

p Mn h.Mn /

(2.3.26) ! 0, we

Therefore, choosing a sequence ¹Mnº such that Mn ! 1 and obtain Z 1 Z 1 Z ap p1 .t /dh.t /  pA.f / h.t /t dt D pA.f / h.s 1=p /ds. (2.3.27) a

a

0

50

Chapter 2 Nonlinear second order elliptic equations with measure data

Hence, by (2.3.22),

Z lim

1

a!1 a

.t /dh.t / D 0

and therefore, by (2.3.25) and (2.3.26), Z 1 h.t /d.t /  lim inf h.a/.a/ D 0.  lim a!1 a

a!1

This proves (2.3.24). Inequality (2.3.26) and (2.3.27) imply that the rate of convergence depends only on h, N , p, and A.f /. p

Now suppose that ¹wm º is a bounded sequence in Lw .; /. The previous assertion implies that there exists M > 0 such that Z h.jwm j/d  < =2. jwm j>M

Put ı D =.2h.M //. Then for every measurable set A  : Z Z .A/ < ı H) .h ı wm /d   h.M /ı C A

jwm j>M

h.jwm j/d  < .

Thus ¹h ı wm º is uniformly integrable. Therefore the last statement of the theorem is a consequence of Vitali’s convergence theorem.  For every a > 0 and  2 M.; / put Z m .a/ D

Œ.x/
 d jj.

(2.3.28)

For a set A  M.; / the modulus of tightness is the function MA given by MA .a/ D sup¹m .a/ :  2 Aº 8a > 0.

(2.3.29)

Thus A is tight if and only if MA .a/ ! 0 as a ! 0. Corollary 2.3.5. Let h 2 C.R/ be an odd monotone increasing function. (i)

Assume that, for some ˇ > 1, Z 1

1N

h.r ˇCN /dr < 1.

(2.3.30)

0

Let ¹m º be a sequence of measures in M.@/ such that m *  weakly relative to C.@/

(2.3.31)

51

Section 2.3 Subcritical nonlinearities

and put wm :D P Œm , Then

w :D P Œ.

wm ! w

in L1 ./,

h ı wm ! h ı w

in L1 .;  ˇ /.

(2.3.32)

(ii) Let ¹m º be a bounded, tight sequence of measures in M.; / such that N / m *  weakly relative to C0 .;

(2.3.33)

and put wm D GŒm ,

w D GŒ.

If h satisfies (2.3.30) then (2.3.32) holds. (iii) Let ¹m º satisfy the assumptions of (ii). In addition assume that ¹m º is bounded in M./. Let wm , w be defined as in (ii). If h satisfies

Z

1

h.r

2N N

/dr < 1

(2.3.34)

0

then

wm ! w

in L1./,

h ı wm ! h ı w

in L1./.

(2.3.35) N Cˇ

Proof. (i) By Lemma 2.3.3 – specifically (2.3.16) – ¹wm º is bounded in LwN 1 .;  ˇ /. Therefore, by Theorem 2.3.4 and (2.3.30), ¹h ı wm º is uniformly integrable in L1.;  ˇ /. By Theorem 1.2.5, wm ! w in L1 ./. By taking a subsequence we may assume that wm ! w a.e. in , so that h ı wm ! h ı w a.e. Therefore, by Vitali’s theorem, h ı wm ! h ı w

in L1.;  ˇ /.

As the limit does not depend on the subsequence it follows that the whole sequence converges. (ii) is proved in exactly the same way except that instead of (2.3.16) we use (2.3.15) (ii). N

N 2 (iii) By (2.3.15), as ¹m º is bounded in M./, ¹wm º is bounded in Lw ./. Consequently, by Theorem 2.3.4 and (2.3.34), ¹h ı wm º is uniformly integrable in L1./. By Theorem 1.2.5 (ii), wm ! w in L1 ./. Therefore by the same argument as before we obtain,  h ı wm ! h ı w in L1./.

52

Chapter 2 Nonlinear second order elliptic equations with measure data

2.3.4 Weak continuity of Sg . Theorem 2.3.6. Let N  3 and g 2 G0. Suppose that there exists h 2 G0, independent of the space variable, such that g.x, t /sign t  h.t /sign t

8.x, t / 2  R.

(2.3.36)

8y 2 , ˛ 2 R

(2.3.37)

If h ı .˛G ., y// 2 L1.; / g

then M.; / D M ./ and

g S

is g-weakly continuous on M.; /.

Proof. Let y 2  and put Dy :D B.y/=2 .y/. In view of (1.1.6) there exist i .y/ such that

1jx  yj2N  G.x, y/  2 jx  yj2N 8x 2 Dy . Since h is non-decreasing and G., y/ is bounded outside Dy it follows that (2.3.37) is equivalent to Z a 2N 2N .h.r N / C h .r N //dr < 1 8a > 0. (2.3.38) 0

Incidentally, this observation implies that if (2.3.37) holds for ˛ D ˙1 then it holds for every ˛ 2 R. Condition (2.3.37) implies that every measure of the form ˛ıy , ˛ 2 R, y 2  is g-admissible. Therefore, by Corollary 2.2.6, ˛ıy 2 Mg ./

8y 2 .

(2.3.39)

Next we prove: Assertion 1. Every measure  2 D./ is g-admissible. Let k X ˛i ıyi , yi 2 , ˛i 2 R, i D 1, : : : , k, D i D1

We may assume that y1 , : : : , yk are distinct points. The solution of jj D v which vanishes on @ is given by vD

k X

j˛i jG., yi /.

i D1

Let ˛ :D

k X

j˛i j,

i D1

:D

1 min min.jyi  yj j, .yi /, .yj //. 4 1i <j k

Then, for x 2 B .yj /, G.x, yj /  G.x, yi / i D 1, : : : , k.

(2.3.40)

53

Section 2.3 Subcritical nonlinearities

Therefore, for j D 1, : : : , k, jv.x/j  ˛G.x, yj /

8x 2 B .yj /

and consequently g ı jvj 2 L1 .; /. This proves the assertion which, in turn, implies D./  Mg ./. Assertion 2. Suppose that ¹m º  Mg ./ is a tight sequence in M.; / which N / to a measure . In addition suppose that, for converges weakly relative to C0.; each m, m is a positive g-admissible measure. Then  2 Mg ./ and g g .m / ! S ./ S

g g g ı .S .m // ! g ı .S .//

in L1 ./,

in L1 .; /. (2.3.41)

We first consider the case where, in addition to the above assumptions, ¹mº is bounded in M./. Put

g .m /, vm :D S

um :D GŒm ,

(2.3.42)

u D GŒ.

By (2.3.38) and Corollary 2.3.5 (iii), um ! u

in L1 ./,

h ı um ! h ı u,

h ı um ! h ı u

in L1 ./.

(2.3.43)

Since vm  um ,

g ı vm  g ı um  h ı um

it follows that ¹g ı vm º is bounded in L1 ./ and consequently ¹vm º is bounded in M./. Therefore ¹vm º is bounded in W 1,p ./, p 2 .1, N=.N 1// and consequently there exists a subsequence (still denoted ¹vm º) converging in L1./ and pointwise a.e. to a function v. Hence (2.3.43) implies that vm ! v in L1./,

g ı vm ! g ı v

in L1 ./.

(2.3.44)

g ./ D v. Since the limit v does not depend These facts imply that  2 Mg ./ and S on the subsequence we conclude that the full sequence ¹vm º satisfies (2.3.44). Next we drop assumption (2.3.42). Let ¹k º be an exhaustion of . Since ¹m º is tight in M.; /, for every > 0 there exists k. / such that Z  dm < 8m 2 N, k  k. /. (2.3.45) nk

Put m,k D m 1k ,

k D 1k ,

g

vm,k D S .m,k /.

54

Chapter 2 Nonlinear second order elliptic equations with measure data

As ¹m,k º is bounded in M./ and m,k ! k , the previous part of the proof implies that there exists wk 2 L1 ./ such that vm,k ! wk in L1./,

g ı vm,k ! g ı wk

in L1./

(2.3.46)

g .k /. and k 2 Mg , wk D S Since ¹k º is monotone increasing and k "  2 M.; /,

k 2 Mg H)  2 Mg and

g

wk " v D S ./ 2 L1 ./,

g ı wk " g ı v 2 L1.; /.

(2.3.47)

g ı vm,k " g ı vm

(2.3.48)

Similarly, since m,k " m 2 M.; /, g

vm,k " vm D S .m / in L1./,

in L1 ./.

By Proposition 2.2.1 and (2.3.45),     vm,k  vm  1 C g ı vm,k  g ı vm  1 L ./ L .;/    C m,k  m 

M ./

 C (2.3.49)

for every m 2 N, k  k. /, where C is a constant depending only on  and g. Hence, for all m, k as above,     kvm  vkL1./  wk  vm,k  C vm  vm,k  C kv  wk k   (2.3.50)  wk  vm,k  1 C kv  wk k 1 C C L ./

L ./

and kg ı vm  g ı vkL1.;/    g ı wk  g ı vm,k 

L1 .;/

C kg ı v  g ı wk kL1 .;/ C C . (2.3.51)

By (2.3.47) there exists k 0 . /  k. / such that kv  wk kL1./ C kg ı v  g ı wk kL1 .;/ 

8k  k 0 . /.

Keeping k fixed, k  k 0 . /, (2.3.46), (2.3.50) and (2.3.51) yield   lim sup kvm  vkL1./  lim wk  vm,k L1./ C .1 C C / D .1 C C / m!1

m!1

and lim sup kg ı vm  g ı vkL1.;/ m!1    lim g ı wk  g ı vm,k L1.;/ C .1 C C / D .1 C C / . m!1

This completes the proof of Assertion 2.

55

Section 2.3 Subcritical nonlinearities

Assertion 3. removed.

The previous assertion remains valid if the assumption m  0 is

Since m is g-admissible, the same is true for jm j. By extracting a subsequence if necessary, we may assume that the sequence ¹.m /C º (resp. ¹.m / º) converges weakly in M.; / to  C (resp.   ). Then  D  C    and, as  ˙  0 it follows that  ˙  ˙ , and hence  :D 1 C 2  jj. Put g

vN m :D S .jm j/,

g

 vN m :D S .jm j/. 

By the previous part of the proof  2 Mg \ Mg and g ./ vN m ! S g

 ! S ./ vN m

g in L1./, g ı .vN m // ! g ı .S .// g

in L1.; /,

 in L1./, g  ı .vN m // ! g  ı .S .// in L1.; /.

(2.3.52)

Since jj   , it follows that  2 Mg . Furthermore, as  /, jvm j  max.vN m , vN m

it follows that ¹vm º is uniformly integrable in L1 ./ while ¹g ı vm º is uniformly integrable in L1 .; /. The sequence ¹vm º is bounded in M.; /. Therefore ¹vm º 1,p is bounded in Wloc ./ for some p > 1. Consequently, by extracting a further subsequence we may assume that ¹vm º converges a.e. This fact and the uniform integrability imply that (2.3.41) holds for the subsequence. But, as the limit is independent of the subsequence, we conclude that (2.3.41) holds for the entire sequence. Completion of proof. Every measure  2 M.; / is the weak limit of a sequence ¹mº  D./. Therefore, combining Assertions 1 and 3 we conclude that Mg ./ D M.; /. Further, if ¹m º is a tight sequence in M.; /, there exists a tight sequence ¹m º in D./ such that N / m  m * 0 relative to C0 .; and kS g .m /  S g .m /kL1./ C kg ı S g .m /  g ı S g .m /kL1.;/ ! 0. Therefore, if ¹m º is a tight sequence such that m *  weakly in M.; / then m * . As (2.3.41) holds with respect to ¹mº it follows that the same is true with  respect to ¹m º.

56

Chapter 2 Nonlinear second order elliptic equations with measure data

g 2.3.5 Weak continuity of S@ .

In the next theorem we derive a condition for subcriticality relative to boundary data. The proof is essentially the same as in the case of problems with interior data. Theorem 2.3.7. Let N  3 and let g 2 G0 . Suppose that there exists h 2 G0 independent of the space variable and ˇ > 2 such that g.x, t /sign t   ˇ .x/h.t /sign t If

8.x, t / 2  R.

(2.3.53)

h ı .˛P ., y// 2 L1.;  ˇ C1 / 8y 2 @, ˛ 2 R

(2.3.54)

g

then M.@/ D M .@/ and

g S@

is g-weakly continuous on M.@/.

Proof. First we show that condition (2.3.54) is equivalent to Z a   1N 1N h.r N CˇC1 / C h .r N CˇC1 / dr < 1 8a > 0.

(2.3.55)

0

Fix y in @ and put r D jx  yj. Consider the conical set CR D ¹x 2  \ BR .y/ : r < 2.x/º. We assume that R is sufficiently small so that CNR   [ ¹yº. Estimate (1.1.7) implies that Z R Z 1Cˇ h.P .x, y//.x/ dx < 1 ” h.r 1N /r N Cˇ dr < 1 CR

and

0

Z

Z 1Cˇ

CR

h.P .x, y//.x/

R

dx < 1 ”

h .r 1N /r N Cˇ dr < 1.

0

This proves the equivalence stated above. Condition (2.3.54) implies that the measure ˛ıy is admissible for every ˛ 2 R, y 2 @. Hence, by Corollary 2.2.6, ˛ıy 2 Mg .@/

8˛ 2 R,

8y 2 @.

(2.3.56)

By the same argument as in the proof of Theorem 2.3.6, we conclude that every measure in D.@/ (= set of linear combinations of Dirac measures supported on @) is admissible. Corresponding to Assertion 2 we have, Assertion 20 . Suppose that ¹m º  Mg .@/ is a sequence converging weakly relative to C.@/ to a measure . In addition suppose that, for each m, m is a gadmissible measure. Then  2 Mg .@/ and g g S@ .m / ! S@ ./

in L1./,

g g g ı .S@ .m // ! g ı .S@ .// in L1.; /. (2.3.57)

57

Section 2.3 Subcritical nonlinearities

We first consider the case where, in addition to the above assumptions, m  0. Put

g

vm :D S@ .m /,

um :D P Œm ,

(2.3.58) u D P Œ.

By (2.3.55) and Corollary 2.3.5 (i), um ! u in L1 ./, h ı um ! h ı u,

h ı um ! h ı u in L1.;  1Cˇ /.

Since vm  um ,

(2.3.59)

g ı vm  g ı um   ˇ h ı um

it follows that ¹gıvmº is bounded in L1.; / and consequently ¹vm º is bounded in 1,p M./. Therefore ¹vm º is bounded in Wloc ./, p 2 .1, N=.N 1// and consequently there exists a subsequence (still denoted ¹vm º) converging pointwise a.e. to a function v. Hence (2.3.59) implies that vm ! v in L1./,

g ı vm ! g ı v in L1.; /.

(2.3.60)

g S@ ./

D v. Since the limit v does not These facts imply that  2 Mg .@/ and depend on the subsequence we conclude that the full sequence ¹vm º satisfies (2.3.60). This result holds for g  as well. The extension to a sequence of signed measures ¹m º is obtained in the same way as in the proof of Assertion 3. Given a measure  2 M.@/ we approximate it by a sequence ¹mº  D.@/. The previous assertion implies that Mg .@/ D M.@/. The proof is completed in the same way as for Theorem 2.3.6.  Examples. 

Let g 2 G0 satisfy jg.x, t /j  c.x/ˇ jt jq ,

ˇ > 2, q > 1 8.x, t / 2  R.

(2.3.61)

If

N C1Cˇ (2.3.62) N 1 g then g is subcritical relative to boundary data and S@ is g-weakly continuous. 1


Let g 2 G0 satisfy

jg.x, t /j  cjt jq

If

8.x, t / 2  R.

(2.3.63)

N (2.3.64) N 2 g then g is subcritical relative to interior data and S is g-weakly continuous. 1
58

Chapter 2 Nonlinear second order elliptic equations with measure data

The next result provides a different kind of sufficient condition for subcriticality and weak continuity. Theorem 2.3.8. Let g 2 G0 such that, for every x 2 , jg.x, /j is an even, convex function. (i) Assume that there exists a number > 0 and a monotone increasing function on .0, 1/ such that kg ı .ˇG .., y//kL1C  .b/ 

8y 2 , ˇ 2 Œb, b.

(2.3.65)

g

Then Mg ./ D M.; / and S is g-weakly continuous on M.; /. (ii) Assume that there exists a number > 0 and a monotone increasing function as above such that kg ı .ˇP .., y//kL1C < .b/ 

8y 2 @, ˇ 2 Œb, b.

(2.3.66)

g is g-weakly continuous on M.@/. Then Mg .@/ D M.@/ and S@

Proof. We prove the first assertion. The second is proved in a similar way. Denote gQ :D jgj1C sign t . If  2 M ./ is gQ admissible then it is g admissible. The conditions on g imply that every measure in D./ is gQ admissible. For every M > 0 there exists a constant c.M / such that  2 D./, jj./  M H) kg ı GŒkL1C ./  c.M /. 

(2.3.67)

This is a consequence of (2.3.65) and the convexity of g. If ¹nº is a sequence in D./ bounded in M./ then, by (2.3.67), ¹g ı GŒjn jº is uniformly integrable in L1.; /.

(2.3.68)

Let un :D S g .n /. Since jun j  GŒjn j, ¹g ı un º is uniformly integrable in L1.; /. By a standard argument, if n *  weakly in M./ then un ! u :D S g ./

in L1./,

g ı un ! g ı u

in L1.; /.

(2.3.69)

By the argument given in the proof of Theorem 2.3.6 – specifically, in the proof of Assertion 2 – one verifies that (2.3.69) remains valid in the case where ¹n º is a sequence in D./, tight in M.; /, such that n *  weakly in M.; /. The completion of the proof is again as in Theorem 2.3.6. 

Section 2.4 The structure of Mg

59

2.4 The structure of Mg In this section we describe properties of Mg under additional assumptions on the nonlinearity g. Definition 2.4.1. Let g 2 G0 ./. Suppose that there exists a function a 2 L1 .; / and non-decreasing functions bi : RC 7! RC, i D 1, 2, such that jg.x, c C t /j  b1.jcj/jg.x, t /j C b2.jcj/a.x/

(2.4.1)

for every t , c 2 R and every x 2 . Then we say that g satisfies the translation subadditivity condition. The family of nonlinearities g 2 G0 possessing this property will be denoted by G1./. Remark 2.4. Let g be in G1 and let u be a measurable function such that g ı u 2 L1.; /. Then g ı .˛ C u/ 2 L1.; / 8˛ 2 L1 ./. (2.4.2) If gu is defined by gu .x, t / :D g.x, t C u.x//  g.x, u.x//

8.x, t / 2  R

(2.4.3)

then gu 2 G0. Note that condition (2.4.1) is satisfied for instance by the function g.t / D expt 1. Thus (2.4.1) is more general than the 2 condition (see (2.4.6) below). Theorem 2.4.2. Suppose that g 2 G1 ./ and that ., / is a g-good pair of measures. If f 2 L1 .; / and h 2 L1.@/ then the problem u C g ı u D  C f in ,

uDCh

on @

(2.4.4)

has a weak solution. Proof. Let u be the weak solution of the problem u C g ı u D  in ,

tr@ u D 

and let gu be defined by (2.4.3). By Proposition 2.1.2, the problem z C gu ı z D f in ,

zDh

on @

(2.4.5)

possesses an L1 weak solution z. In particular, g ı.uCz/ 2 L1.; / and z 2 L1 ./. Thus, if v :D z C u then v 2 L1 ./, g ı v 2 L1.; / and v C u C g ı v  g ı u D f in ,

v D hC

on @.

The above equation is equivalent to v C g ı v D f C 

in .



60

Chapter 2 Nonlinear second order elliptic equations with measure data

Corollary 2.4.3. Let g 2 G1 . Let  2 Mg ./ and let s be the singular part of  relative to Lebesgue measure. Then  2 Mg ./ if and only if s 2 Mg ./. 

Proof. This is an immediate consequence of Theorem 2.4.2.

Definition 2.4.4. Let g 2 G0. We say that g satisfies the 2 condition if there exists a constant c such that g.x, a C b/  c.g.x, a/ C g.x, b// 8x 2 , a > 0 b > 0.

(2.4.6)

We denote G2 D ¹g 2 G0 : g satisfies (2.4.6)º. Theorem 2.4.5. Let g 2 G0 and let  2 M ./ and  2 M.@/ be non-negative measures. Then: N H)  2 Mg ./,  2 Mg .@/. ., / 2 Mg ./

(2.4.7)

If, in addition, g 2 G2 then N (H  2 Mg ./,  2 Mg .@/. ., / 2 Mg ./

(2.4.8)

N and let U be the solution of the problem Proof. Suppose that ., / 2 Mg ./ u C g ı u D  in ,

uD

on @.

(2.4.9)

Then U  0 and U is a supersolution of each of the following problems: .a/ .a/

u C g ı u D  in , u C g ı u D 0 in ,

uD0 uD

on @, on @.

(2.4.10)

Clearly u 0 is a subsolution for each of these problems. By Theorem 2.2.4, (2.4.10) (a) and (b) possess weak solutions. Conversely, assume that (2.4.10) (a) and (b) possess weak solutions u and v respectively. By Theorem 2.4.2  C g ı v 2 Mg ./. Therefore there exists a weak solution w of the problem w C g ı w D  C g ı v

in ,

tr@ w D 0.

Assuming that g 2 G2, g ı .w C v/ 2 L1 .; / and w C v is a supersolution of (2.4.9): .w C v/ C g ı .w C v/ D w C g ı .w C v/  g ı v   tr@ .w C v/ D .

in ,

On the other hand v is a subsolution. Therefore, by Theorem 2.2.4, (2.4.9) possesses a solution. 

Section 2.4 The structure of Mg

61

Proposition 2.4.6. Suppose that g 2 G2 and that g.x, / is super-additive: g.x, a C b/  g.x, a/ C g.x, b/

8x 2 ,

8a, b 2 RC .

(2.4.11)

Let ,  be two non-negative measures in Mg ./. Then (i)  C  2 Mg ./. (ii) j  j 2 Mg ./. (iii) If f 2 L1.; d/ then f d 2 Mg ./. (iv) The space Mga ./ D ¹   : ,  2 Mg ./, ,   0º g

is a linear space and Ma ./  Mg ./. The parallel assertions with respect to Mg .@/ are also valid. In particular Mga .@/ D ¹   : ,  2 Mg .@/, ,   0º  Mg .@/, g

Ma .@/ is a linear space and it is closed relative to the operator  7! jj. Finally, these assertions remain valid for the space N D Mga .@/ Mga ./. Mga ./ Proof. Suppose that ,  2 M./ and ,  2 M.@/ are non-negative and that ., / N Let u and v be the weak solutions of and ., / are in Mg ./. .a/ .b/

 u C g ı u D   v C g ı v D 

tr@ u D  tr@ u D .

in !, in !,

(2.4.12)

Since g 2 G2, g ı .u C v/ 2 L1 .; / and the super-additivity implies that u C v is a supersolution of the equation w C g.w/ D  C . Clearly tr@ .u C v/ D  C . Thus u C v is a supersolution of the problem w C g.w/ D  C 

in ,

trbdw .u C v/ D  C .

Since u (as well as v) is a subsolution of this problem it follows, by Theorem 2.2.4, that this problem has a solution w and max.u, v/  w  u C v.

(2.4.13)

N Choosing  D  D 0 we In particular we conclude that . C ,  C / 2 Mg ./. conclude that  C  2 Mg .@/. Similarly  C  2 Mg ./. Further, by Proposition 2.2.1, .u  v/C is a weak subsolution of the problem Z C g ı Z D .  /C

in ,

tr@ Z D .  /C .

Since u is a weak supersolution of this problem, Theorem 2.2.7 implies that there exists a weak solution of the problem w C g ı w D .  /C

in ,

tr@ w D .

(2.4.14)

62

Chapter 2 Nonlinear second order elliptic equations with measure data

N In the same way we obtain ..  /C, / 2 This implies that ..  /C , / 2 Mg ./. g N N This of M ./ and finally, by the first part of the proof .j  j,  C / 2 Mg ./. g course implies that    2 M ./. N and finally, by TheoA similar argument shows that . C , j  j/ 2 Mg ./ rem 2.4.5, N .j  j, j  j/ 2 Mg ./. By the first part of the proof, k 2 Mg ./ for every natural number k. This implies that ˛ 2 Mg ./ for every ˛ 2 RC. Similarly, ˛ 2 Mg .@/ for every positive ˛. g If  is an arbitrary measure in Ma ./ then, as shown above, jj 2 Mg ./. Thereg fore j˛j 2 M ./ and hence ˛ 2 Mg ./ for any real ˛. g If ,  are two arbitrary measures in Ma ./ then jjCjj 2 Mg ./ and therefore g  C  2 Mg ./. Therefore we conclude that Ma ./ is a linear space. A similar g argument shows that Ma .@/ is a linear space and both spaces are closed with respect g N follows to the action  7! jj. The corresponding conclusion with respect to Ma ./ from the above. It remains to prove assertion (iii). If  is a measure in Mga ./ (not necessarily positive) then jj is in the same space. If f 2 L1 ., jj/ then f d is dominated by

jj where is the L1 norm of f . Since jj 2 Mg ./ it follows that f d is in this space. In the general case, it is sufficient to prove that jf jd jj 2 Mg ./. This is verified by approximating jf j by the sequence ¹hnº where hn D min.jf j, n/. The  corresponding result for Mg .@/ is proved in the same way. Remark 2.5. Under the assumptions of Proposition 2.4.6, it is possible that g.u/ 2 L1.; / but g.juj/ is not in this space. Therefore, it is possible that  2 Mg but jj 62 Mg . In view of this, it is useful to record the following. Proposition 2.4.7. Suppose that g 2 G2, g satisfies (2.4.11) and g.x, / is odd. Then N H) .jj, jj/ 2 Mg ./ N ., / 2 Mg ./

(2.4.15)

and Mg ./, Mg .@/ are linear spaces. N Let u be the weak solution of the boundary Proof. Suppose that ., / 2 Mg ./. value problem u C g ı u D  in , uD on @. Let w ˙ be the solution of w ˙ C g ı u˙ D ˙ w ˙ D ˙

in , on @.

The existence of these solutions is guaranteed by the fact that g ı juj 2 L1 .

63

Section 2.5 Remarks on unbounded domains

Now, u D w C  w  and consequently, uC  w C, u  w . Therefore, if w :D C w  then w C g ı w  jj in ,

wC

in the sense of distributions and tr@ w D jj. By Lemma 1.5.8 it follows that there exists a positive measure  2 M.; / such that w C g ı w D jj C 

in .

By Proposition 1.3.7 w is a weak solution of the boundary value problem w C g ı w D jj C  w D jj

in , on @.

N and consequently .jj, jj/ 2 Mg ./. N Thus .jj C , jj/ 2 Mg ./

2.5 Remarks on unbounded domains We briefly discuss extensions of some of the previous results to a class of unbounded domains. Definition 2.5.1. A possibly unbounded domain  is uniformly of class C 2 if it satisfies the following conditions: (i) There exists r0 > 0 such that, for every X 2 @, there exists a set of coordinates  D  X and a function FX as in Definition 1.1.1. (ii) The set ¹FX : X 2 @º can be chosen so that k@kC 2 :D sup¹kFX kC 2 .BN r

0 .0//

: X 2 @º < 1

and there exists 2 C.0, 1/ such that D 2 FX has modulus of continuity for every X 2 @. Remark 2.6. If  is a bounded domain satisfying (i), then there always exists a family ¹FX : X 2 @º satisfying (ii). We denote by M0 .@/ the family of set functions on B.@/ such that, for every compact F  @, 1F is a finite measure. Similarly, M0 ./ denotes the family of set functions  on B./ such that, for every R > 0, 1R 2 M .R /,

R :D  \ BR .0/.

64

Chapter 2 Nonlinear second order elliptic equations with measure data

Further we denote by .L1/0 ./ (resp. .L1 /0 ./) the set of functions u 2 L1loc ./ such N denotes the that u 2 L1 .R / (resp. u 2 L1 .R /) for every R > 0. Finally .C02 /0 ./ N consisting of functions with bounded support. subset of C02 ./ 1,p If u 2 Wloc ./, we shall say that it has an M-boundary trace  2 M0 .@/ if, for every bounded relatively open set A  @, trA u D 1A . The definition of a weak solution is extended as follows: Definition 2.5.2. Assume that  2 M0 ./,

 2 M0 .@/.

(2.5.1)

A function u 2 .L1/0 ./ is a weak solution of the problem (1.2.4) if it satisfies (1.2.5) N for every 2 .C02 /0 ./. Definition 2.1.1 is extended to unbounded domains in the same way. If  is unbounded problem (1.2.4) may not have a solution and uniqueness fails. However, existence can be established under various conditions on the data and uniqueness may hold if additional restrictions are imposed on the solution. Here is a simple example of an existence result. Proposition 2.5.3. Assume that  is uniformly of class C 2 . Suppose that  2 M0 ./ and  2 M0 .@/. In addition assume that Z Z 2N jxj d.x/ < 1, jxj1N d.x/ < 1. (2.5.2) \Œjxj>1

@\Œjxj>1

Then problem (1.2.4) possesses a solution. The proof is based on Theorem 1.2.2 and estimates for the Green and Poisson kernels near the boundary (using the uniform smoothness of the boundary) and in the interior (using (2.5.2)). The proposition will not be used in the following and the proof is omitted. An existence and uniqueness result for some nonlinear problems – with  D 0 and general boundary data – will be established in Chapter 3. We note that Proposition 1.3.7 – with obvious modifications – remains valid with respect to weak solutions in unbounded domains in the sense of Definition 2.5.2. The same applies to Theorem 2.2.7, whose proof is based on this proposition, and also to Corollary 2.2.8 and Theorem 2.2.9.

2.6 Notes The study of boundary value problems for semilinear elliptic PDE in an L1 framework was pursued by H. Brezis and coworkers in the early 1970s. In a set of lecture notes

Section 2.6 Notes

65

H. Brezis [21] introduced the notion of a weak L1 solution (see Definition 1.1.2) for problems with L1 data and derived a priori estimates which were then used to establish results of existence and uniqueness as in H. Brezis and W. Strauss [26]. A comprehensive study of a family of semilinear elliptic boundary value problem with measure data was carried out by H. Brezis and Ph. Benilan in the mid-1970s. This seminal work was fully reported only in 2003 (see [14]), but some results appeared in [22]. Among other results this paper included a sufficient condition for the subcriticality of g g and the g-weak continuity of S (see Theorem 2.3.6) which was based on estimates p in weak L spaces, also known as Marcinkiewicz spaces. This method was later applied by A. Gmira and L. Véron [54] in order to derive a similar result for the operator g S@ in the case that the nonlinearity g is dominated by a function independent of the spatial variable. The more general result presented in Theorem 2.3.7 was obtained by M. Marcus [72]. In this paper it is also shown that – assuming (3.1.28) holds for every positive solution – the subcriticality condition is necessary as well as sufficient. The results stated in Theorems 2.2.4, 2.2.7 and 2.2.9 are taken from an unpublished set of notes by M. Marcus [73]. For the general class of nonlinearities discussed here they seem to be new. Related results can be found in various other papers. Theorem 2.2.4 was independently obtained by M. Montenegro and A. Ponce [90]. In the case g.t / D jt jq sign t , the results presented in Section 2.4 were obtained by M. Marcus and L. Véron [78]. In the present form, they were established in [73].

Chapter 3

The boundary trace and associated boundary value problems 3.1 The boundary trace In the present section the default assumption on the domain is:  is bounded of class C 2. However, as we shall point out, many of the results also apply to unbounded domains that are uniformly of class C 2 . In the latter case we modify the definition of G0./ as follows: g 2 G0 ./

if, for every C 2 bounded domain 0  , g 2 G0.0 /.

3.1.1 Moderate solutions Let g 2 G0. A solution of equation u C g ı u D 0

(3.1.1)

is a function u 2 L1loc ./ such that g ı u 2 L1loc ./ and satisfies the equation in the sense of distributions, i.e., Z Z  u dx C .g ı u/ dx D 0, 



for every 2 Cc1 ./. 1,p We recall that every solution of (3.1.1) is in the space Wloc for every p 2 Œ1, N= 0 0 2 .N  1//. Consequently, if  b  and  is of class C then u possesses a trace on @0 given by a function h 2 Lp .@0 /. In fact, for every 1 < p < N=.N  1/, h is in 0 W 1=p ,p .@0 /, p1 C p10 D 1. Definition 3.1.1. A solution u of (3.1.1) is moderate if there exists a positive harmonic function U such that juj  U in . Theorem 3.1.2. A solution u of (3.1.1) is moderate if and only if it possesses an Mboundary trace. Furthermore, if u is moderate then g ı juj 2 L1.; / and u is a weak solution (as in Definition 2.1.1) of the boundary value problem u C g ı u D 0 in , u D  on @ where  D tr u.

(3.1.2)

67

Section 3.1 The boundary trace

Proof. First assume that u is a positive moderate solution of (3.1.1). Let U be a harmonic function such that u  U . By the theorem of Herglotz, U has an M-boundary trace. Therefore Z lim U dS < 1. ˇ !0 †ˇ

(For the notation see (1.3.1).) It follows that Z sup u dS < 1 and u 2 L1./.

(3.1.3)

†ˇ

This implies that there exists a sequence ¹ˇn º decreasing to zero and a measure  2 M.@/ such that Z Z lim

n!1 † ˇn

uh dS !

h d,

N Since u is a solution of (3.1.1), for every h 2 C./. Z Z Z u dx C .g ı u/ dx D  Dˇ

Dˇ

(3.1.4)

@

†ˇ

u@n dS

(3.1.5)

N let 'n and ' be as in for every 2 C02.Dˇ / and every ˇ 2 .0, ˇ0 /. Given h 2 C 2 ./, Lemma 1.3.4. Consider (3.1.5) with ˇ D ˇn and D 'n . By (3.1.3)–(3.1.4) the first and third terms in (3.1.5) converge as n ! 1. Therefore the second term converges and we obtain, Z Z Z  u' dx C .g ı u/' dx D @n' d. (3.1.6) 



@

In particular, if we choose h D 1, the function ' in Lemma 1.3.4 could be taken to be identical to  for  < ˇ0 =2. Therefore g ıu 2 L1.; /. By Corollary 2.2.8, a positive solution of (3.1.1) such that g ı u 2 L1.; / has an M-boundary trace, which in the present case is the measure  mentioned above. Next assume that u is moderate (not necessarily positive) and juj  U . Then juj is a subsolution of (3.1.1) and U is a supersolution. The smallest solution above juj, say w, is also dominated by U . By the previous part of the proof g ı w 2 L1.; /. Therefore g ı juj 2 L1.; / and Z sup juj dS < 1. †ˇ

We extract a sequence ¹ˇn º such that (3.1.4) holds and, as in the first part of the proof, we employ (3.1.5) with ˇ D ˇn and ' D 'n . In the present case, the left-hand side of (3.1.5) converges and consequently the right-hand side converges. The limit on the left-hand side is independent of the sequence ¹ˇn º; therefore the measure  is independent of the sequence and it is the M-boundary trace of u. The last assertion is a consequence of Proposition 1.3.7. 

68

Chapter 3 The boundary trace and associated boundary value problems

Next we provide a local version of the last result. Definition 3.1.3. Let u be a solution of (3.1.1). Let Q be a bounded open set such that Q \ @ ¤ ;. Suppose that, Z sup jujdS < 1. (3.1.7) 0<ˇ <ˇ0 †ˇ \Q

Then we say that u is moderate in Q \ . Proposition 3.1.4. Let Q be as in Definition 3.1.3. Suppose that u is a solution of (3.1.1), moderate in Q \ . Then there exists  2 M.Q \ @/ such that  is the M-boundary trace of u on Q \ @. Furthermore, for every open set Q0 b Q, g ı juj 2 L1 .Q0 \ /,

.x/ D dist .x, @/.

Proof. First assume that u is positive. Condition (3.1.7) implies that u 2 L1.Q \ / and that there exists a sequence ¹ˇnº decreasing to zero and a measure  2 M.Q\@/ such that Z Z lim uh dS ! h d, (3.1.8) n!1 † \Q ˇn

@\Q

N for every h 2 C./. 2 N N and let 'n and ' be Let h 2 C ./ be a function such that supp h  Q \  N constructed as in the proof of Lemma 1.3.5 so that supp 'n  Q \ . Consider (3.1.5) with ˇ D ˇn and D 'n . When n ! 1, the first term converges because u 2 L1.Q \ / and the third term converges by (3.1.8). Therefore the second term converges and we obtain (3.1.6) with ' as above. N such that h D 1 in Let Q0 b Q be such that Q0 \ @ ¤ ;. Choose h 2 C 2 ./ 0 N Q \ , h  0 and supp h  Q \ . Then the construction in Lemma 1.3.5 allows us to choose ' non-negative such that '.x/ D .x/ 8x 2 Q0 : .x/ < ˇ0 =2. Therefore g ı u 2 L1 . \ Q0 /. Let Q b Q0 be an open set such that Q \ @ ¤ ; and let D be a C 2 domain such that N \ Q0 .  \ Q  DN   By Corollary 2.2.8 u possesses an M-boundary trace on @D. In view of (3.1.8), this boundary trace coincides with  on @D \ @. We turn to the general case: u is moderate in Q \  but not necessarily positive. Then juj is a subsolution of (3.1.1). Let D be a C 2 domain with the properties described above. Denote @1D D @D \ , @2D D @D \ @.

69

Section 3.1 The boundary trace 1,p

Since u 2 Wloc ./, u has a (Sobolev) trace h1 on @1D such that h1 2 L1.F / for every compact set F  @1D. As u 2 L1.Q \ / we may choose D so that h1 2 L1.@1 D/. Let Dˇ D D \ Dˇ and let wˇ be the weak solution of the boundary value problem: w C g ı w D 0

in Dˇ ,

w D juj on @.Dˇ /. Then,     wˇ  1  C g ı wˇ  1 L .D / L ˇ

 .Dˇ /  ˇ

 c kukL1 .@D / ˇ



Z

 c kukL1.@1 D/ C

†ˇ \Q

 jujdS .

(3.1.9)

Here ˇ .x/ D dist .x, @Dˇ /. The family ¹wˇ º increases as ˇ # 0. Therefore w D limˇ !0 wˇ is well defined. Since the right-hand side of (3.1.9) is bounded we conclude that w 2 L1.D/,

g ı w 2 L1@D .D/

where @D .x/ D dist .x, @D/. Clearly w is the smallest solution of (3.1.1) in D dominating juj. It follows that u 2 L1D .D/ where D .x/ D dist .x, @D/. Consequently u has an M-boundary trace on @D. Therefore the measure  in (3.1.8) is independent of the sequence.  An examination of the previous proof shows that it actually yields a slightly stronger result: Proposition 3.1.5. Let Q be as in Definition 3.1.3. Suppose that u is a solution of (3.1.1) such that Z 1 jujdS < 1. (3.1.10) u 2 L .Q \ / and lim inf ˇ !0

†ˇ \Q

Then the conclusions of Proposition 3.1.4 remain valid. Positive moderate solutions can be characterized as follows: Proposition 3.1.6. Let u be a positive solution of (3.1.1). Then u is moderate if and only if g ı u 2 L1.; /. Proof. If gıu 2 L1 .; / then, by Corollary 2.2.8, u has an M-boundary trace. Therefore, by Theorem 3.1.2, u is moderate. The converse is part of Theorem 3.1.2. 

70

Chapter 3 The boundary trace and associated boundary value problems

3.1.2 Positive solutions Here we consider arbitrary positive solutions of (3.1.1), in general non-moderate. As we study positive solutions, the behavior of g on R plays no role. We show that, under various conditions on g, to every positive solution there corresponds a positive Borel measure on @, in general unbounded, which we call a boundary trace. This can be done in more than one way. The aim is to provide a definition of trace such that: (a) for every boundary trace there exists a solution and (b) the solution is uniquely determined by its boundary trace. The present section is devoted to the construction of the boundary trace. The problem of existence and uniqueness is studied in the next section where we restrict ourselves to equations with power nonlinearity in the subcritical case. Motivated by Proposition 3.1.6 we introduce the following: Definition 3.1.7. Let u be a positive solution of (3.1.1). A point y 2 @ is regular relative to u if there exists a neighborhood Q of y such that Z .g ı u/ dx < 1. (3.1.11) Q\

Otherwise we say that y is a singular point relative to u. The set of regular points is denoted by R.u/; the set of singular points is denoted by S.u/. Evidently R.u/ is a (relatively) open subset of @. Theorem 3.1.8. Suppose that g 2 G0 . Let u be a positive solution of (3.1.1). Then: (i) u has an M-boundary trace on R.u/ given by a positive Radon measure . Thus, Z Z hu dS D hd, (3.1.12) lim ˇ !0 †ˇ

@

N vanishing in a neighborhood of S.u/. for every h 2 C./ (ii) A point y 2 @ is singular with respect to u if and only if, for every r > 0, Z u dS D 1. (3.1.13) lim sup ˇ !0

Br .y/\†ˇ

Proof. (i) Let D be a C 2 domain such that D   and .@D \ @/  R.u/. Then g ı u 2 L1 .D/ and consequently, by Proposition 3.1.6, u possesses an M- boundary trace on @D. This implies that u possesses an M-boundary trace on R.u/, i.e., there exists a positive Radon measure  on R.u/ such that (3.1.12) holds. (ii) By negation assume that y 2 S.u/ and Z lim sup u dS < 1. ˇ !0

Br .y/\†ˇ

71

Section 3.1 The boundary trace

Then, for every open set Q such that QN  Br .y/, u is moderate in Q \  (see Definition 3.1.3). By Proposition 3.1.4, g ı u 2 L1 .Q \ /, which contradicts the assumption. On the other hand, if (3.1.13) holds for every r > 0 then, by part (i), y 62 R.u/.  Definition 3.1.9. (a) If h 2 C.R/, h.0/ D 0 and h is non-decreasing, we say that it satisfies the Keller–Osserman condition if, for every a > 0, Z s Z 1 1=2 H.s/ ds < 1, H.s/ D h.t /dt . (3.1.14) a

0

Let g 2 G0. We say that g satisfies the global Keller–Osserman condition if there exists h as above such that h satisfies the Keller–Osserman condition and h.t /  min.jg.x, t /j, jg  .x, t /j/

8x 2 , t 2 RC .

(3.1.15)

(b) We say that g satisfies the local Keller–Osserman condition if, for every domain 0 b  there exists h D h0 2 C.R/ which satisfies the Keller–Osserman condition and h0 .t /  min.jg.x, t /j, jg  .x, t /j/

8x 2 0 , t 2 RC.

The Keller–Osserman inequality (Keller [60] and Osserman [96]). Assume that g 2 G0 and that it satisfies the global Keller–Osserman condition. Then there exists a non-increasing function fg : RC 7! RC tending to infinity at zero and to zero at infinity such that every solution u of (3.1.1) satisfies ju.x/j  fg ..x// 8x 2 .

(3.1.16)

A proof of this inequality is supplied in Chapter 4. Note that if u is a solution of (3.1.1) then, by Kato’s inequality (specifically Corollary 1.5.5). juj satisfies juj C g.x, u/sign u  0. Therefore, (3.1.16) follows from Theorem 4.1.2. As a consequence we have: Suppose that g 2 G0 and satisfies the local Keller– Osserman condition. Then, for every compact set K   there exists a positive constant CK such that every solution u of (3.1.1) satisfies sup ju.x/j  CK .

x2K

(3.1.17)

72

Chapter 3 The boundary trace and associated boundary value problems

Definition 3.1.10. Let z 2 @. We say that there exists a g-barrier at z if there exists r .z/ > 0 such that, for every 0 < r  r .z/, (3.1.1) possesses a solution w D wr ,z in Br .z/ \  such that: N wD0 w 2 C.Br .z/ \ /, lim w.x/ D 1

x!y

on @ \ Br .z/,

8y 2 @Br .z/ \ .

(3.1.18)

We say that g satisfies the global barrier condition if: (i) A g-barrier exists at every point of the boundary. (ii) There exists a number rN > 0 such that r .z/  rN 8z 2 @.  C where C is a constant independent of z. (iii) If w D wrN ,z then kwkC 2 .\B N r=2 N .z// Lemma 3.1.11. Suppose that  is a uniformly C 2 domain (possibly unbounded) and g 2 G0 satisfies the global Keller–Osserman condition in . Then g satisfies the global barrier condition. Proof. Let h be as in Definition 3.1.9 and let r > 0. Let wr denote the maximal solution of the equation w C h.w/ D 0 in Br .0/.

(3.1.19)

The existence of a maximal solution follows from the Keller–Osserman inequality and Lemma 3.2.1. If w is a solution of this equation then jwj is a subsolution. Therefore wr > 0. Evidently lim wr .x/ D 1 8y 2 @Br .0/. x!y

If z 2 @ and r 2 .0, ˇ0 / then w D wr .  z/ is a supersolution of the equation u C g ı u D 0 in Br .z/ \ . If UM denotes the solution of this equation with boundary data UM D 0 on Br .z/ \ @,

UM D M

on @Br .z/ \ 

then UM  wr .  z/. Consequently uz D limM !1 UM vanishes on Br .z/ \ @; obviously uz blows up at @Br .z/\. Therefore uz is a g-barrier at z. We may choose rN :D ˇ0 =2 so that, for every z 2 @, the barrier constructed above exists in BrN .z/\. Since uz  wrN .  z/ in BrN .z/ \  we conclude that there exists C > 0 such that sup¹uz .x/ : x 2 BrN =2 .z/ \ º < C

8z 2 @.

By standard elliptic estimates, this implies: < 1. sup kuz kC 2 .\B N r=2 N .z//

z2@



73

Section 3.1 The boundary trace

Next we establish: Theorem 3.1.12. Assume that g 2 G0 and that, for every z 2 @, there exists a g-barrier at z. Let u be a positive solution of (3.1.1) and suppose that y 2 S.u/. Then, for every r > 0, Z lim u dS D 1. (3.1.20) ˇ !0 Br .y/\†ˇ

Proof. By assumption

Z Br .y/\

.g ı u/ dx D 1

Therefore Proposition 3.1.4 implies that Z udS D 1 lim sup ˇ !0

Br .y/\†ˇ

8r > 0.

8r > 0.

(3.1.21)

(3.1.22)

We have to show that the lim inf of this expression is not finite. Suppose, by negation, that there exist r > 0 and a sequence ¹ˇn º decreasing to zero such that Z udS < 1. (3.1.23) lim n!1 B .y/\† r ˇn

We may and shall assume that r < r .y/ (see Definition 3.1.10). Let wn be the solution of (3.1.1) in Br .y/ \ Dˇn (Dˇ as in (1.3.1)) with boundary trace fn D u on †ˇn \ Br .y/, fn D 0 on @Br .y/ \ Dˇn . Extracting a subsequence if necessary we may assume that ¹wnº converges to a solution w in Br .y/ \ . Note that Z udS. kwnkL1  Br .y/\†ˇn

Therefore, in view of (3.1.23), w 2 L1.Br .y/ \ /. Since r < r .y/ there exists a solution v of (3.1.1) in Br .y/ \  such that v D 1 on @Br .y/ \ ,

v D 0 on @ \ BN r .y/.

Then wn C v is a supersolution of (3.1.1) in Br .y/ \ Dˇn and u  wn C v on the boundary of this domain. Therefore u  wn C v in Br .y/ \ Dˇn and consequently u  w C v. Since v 2 L1.Bs .y/ \ ) for every 0 < s < r and w 2 L1.Br .y/ \ /, it follows that u 2 L1 .Bs .y/ \ / for every such s. In view of Proposition 3.1.5, this fact and (3.1.23) contradict (3.1.21). 

74

Chapter 3 The boundary trace and associated boundary value problems

Definition 3.1.13. Let F be a closed subset of @ and let  be a positive Radon measure on @ n F . Let N be the Borel measure on @ given by ´ .E/ if E  @ n F .E/ N D (3.1.24) 1 if E \ F ¤ ; for every Borel subset of @. Denote by Breg the space of positive Borel measures  N on @ which can be represented by (3.1.24) for a pair ., F / as above. Notation. For  2 Mg .@/ we denote by u ,g the solution of the boundary value problem u C g ı u D 0 in , (3.1.25) u D  on @. Definition 3.1.14. Let g 2 G0 and u be a positive solution of (3.1.1). Let  be the M-boundary trace of u on R.u/ (see Theorem 3.1.8). Then the measure N 2 Breg represented by the pair ., S.u// is called the ‘rough boundary trace’ of u, denoted by Tr u. The measure  is called the regular part of the trace and denoted by Trregu. Remark 3.1.14. For every compact F  R.u/, denote by vF the solution of (3.1.25) with  replaced by 1F . The function V,g :D sup¹vF : F  R.u/, F compactº

(3.1.26)

is a solution of (3.1.1) because vF1 _ vF2 D vF1 [F2 and for every F as above, vF  u. The solution V,g is called the regular component of u. Theorem 3.1.15. Suppose that g.x, t / D .x/ˇ h.t / with ˇ > 2 and h 2 C.R/ an odd and non-decreasing function such that t 7! h.t /=t is non-decreasing for large t . Suppose that h ı .˛P ., y// 2 L1.;  ˇ C1/

8y 2 @, ˛ 2 R.

(3.1.27)

Let u be a positive solution of (3.1.1) and denote Hu .x/ D  ˇ .x/h.u.x//=u.x/

8x 2 .

Assume that, for some constant cu , Hu.x/  cu .x/2

8x 2 .

(3.1.28)

Then, for y 2 @, y 2 S.u/ ” u  u1,y,g :D lim uk,y,g . k!1

The proof of the theorem is based on two lemmas.

(3.1.29)

75

Section 3.1 The boundary trace

Lemma 3.1.16. Assume that v is a positive solution of the equation v C H v D 0

(3.1.30)

N / :D sup H , t 2 .0, ˇ0 / and suppose that where H 2 C 2 ./. Let h.t †t Z ˇ0 N / dt < 1. t h.t

(3.1.31)

0

Then v 2 L1./ and H v 2 L1.; /. Furthermore, v has an M-boundary trace on @. Proof. Let 2 C 2.@/ and put  D .  ˇ/ . For 0 < ˇ < ˇ0 , Z Z .v C H v/dx D v dS  R1 Dˇ

(3.1.32)

†ˇ

Z

where R1 D

†ˇ0

.v  .@v=@n//dS.

Using (3.1.32) with 1 we obtain Z Z    N vdS  c0 v.x/ h..x//.x/ C 1 dx C 1 ,

(3.1.33)

where c0 depends on ˇ0 but not on ˇ. Put Z ˇ0 Z N vdS, !.ˇ/ :D .h.ˇ/ˇ C 1/U.ˇ/dˇ C 1. U.ˇ/ :D

(3.1.34)

†ˇ

Dˇ

†ˇ

ˇ

By (3.1.33) and (3.1.34), U.ˇ/  c1 !.ˇ/.

(3.1.35)

N C 1/U.ˇ/ it follows that Since ! 0 .ˇ/ D .h.ˇ/ˇ N ! 0 .ˇ/=.h.ˇ/ˇ C 1/  c1 !.ˇ/. Then .e A.ˇ /!.ˇ//0  0,

Z where

ˇ

A.ˇ/ D c1

N /t /dt . .1 C h.t

(3.1.36)

(3.1.37)

0

It follows that

e A.ˇ /!.ˇ/  e A.ˇ0/ !.ˇ0 /

8ˇ 2 .0, ˇ0 /

and consequently ! is bounded. By (3.1.34) and (3.1.35) this implies Rthat U is bounded and that v and H v are integrable. Hence, by (3.1.32) limˇ !0 †ˇ v dS exists for every 2 C 2.†/. Since v > 0 it follows that the limit exists for every

2 C.†/. 

76

Chapter 3 The boundary trace and associated boundary value problems

Lemma 3.1.17. Assume that g.x, t / D .x/ˇ h.t /. Under the conditions of the theorem, if y 2 S.u/ then there exists a sequence ¹n º   converging to y such that u.n /.n /N 1 ! 1.

(3.1.38)

Proof. By negation, if the assertion is not valid there exists R 2 .0, 1/ and a constant C such that u.x/ < C.x/1N 8x 2 BR .y/ \ . (3.1.39) As h.t /=t is non-decreasing, it follows that Hu   ˇ

h.C 1N / D C 1  ˇ CN 1h.C 1N / C 1N

in Q :D  \ BR .y/. Thus, fu .t / :D

sup † t \BR .y/

Condition (3.1.27) implies that Z 1

Hu  C 1t ˇ CN 1 h.C t 1N /.

t ˇ CN h.C t 1N /dt < 1

0

(see proof of Theorem 2.3.7). Therefore fu satisfies (3.1.31); by Lemma 3.1.16, u 2 L1.Q/ and  ˇ h.u/ 2 L1 .Q/. Therefore y 2 R.u/, contrary to our assumption.  Proof of Theorem 3.1.15. For every R > 0,

Z lim

t !0 BR .y/\† t

uk,y,g dS D k.

Therefore, the right-hand side of (3.1.29) implies that Z u dS D 1. lim t !0 BR .y/\† t

By Theorem 3.1.8 (i), y 2 S.u/. We turn to the main part of the proof: assuming that y 2 S.u/ we show that the right-hand side of (3.1.29) holds. Let ¹n º be a sequence in  as in Lemma 3.1.17. Put Bn :D B. n /=2 .n /. As u is a positive solution of the equation u C Huu D 0, assumption (3.1.28) and the classical Harnack inequality imply that there exists a constant c, N depending on  and the constant in (3.1.28), such that sup u  cN inf u. Bn

Bn

(3.1.40)

77

Section 3.1 The boundary trace

In the remaining part of the proof, ci denotes a constant depending on  and c. N We may and shall assume that jn j < ˇ0 =4. Denote

n D .n /, n D .n /, Dn D ¹x 2  : .x/ > n º, Vn D Bn \ †n . (Here ./ is defined as in Proposition 1.3.2.) Then by (3.1.40) and (3.1.38) Z u dS  c1 .sup u/ nN 1 ! 1. (3.1.41) bn :D Vn

Bn

Let fn,k be a function on †n given by fn,k D .k=bn /u

in Vn ,

fn,k D 0 in †n n Vn .

(3.1.42)

Then, by (3.1.41), fn,k  kc11 n1N

and

fn,k dS †n * kı0

N weakly relative to C./

(3.1.43)

where dS †n denotes the surface element on †n . Let wn,k denote the solution of the boundary value problem w C g ı w D 0 w D fn,k

in Dn on @Dn.

(3.1.44)

It will be convenient to extend wn,k to  by setting wn,k D 0 outside Dn . The extension will also be denoted by wn,k . Given k > 0 pick n.k/ such that bn  k

in Vn

8n  n.k/.

Then fn,k  u on †n and consequently wn,k  u

in Dn

8n  n.k/.

(3.1.45)

Further, by (3.1.43), P .x, n /  c2 n1N  c3 fn,k .x/=k

8x 2 Vn .

Therefore, by the maximum principle, .k=c3 /P .x, n /  wn,k .x/

8x 2 Dn .

Condition (3.1.27) implies that Z 1 1N h.C t 1CˇCN /dt < 1, 0

(3.1.46)

78

Chapter 3 The boundary trace and associated boundary value problems

(see proof of Theorem 2.3.7). As n ! y it follows that n ! y. Therefore, by Corollary 2.3.5 (i) applied to the sequence ¹ ck3 ı n º, it follows that P ., n / ! P ., y/ in L1./, k k h ı P ., n / ! h ı P ., y/ in L1 .;  ˇ C1/. c3 c3

(3.1.47)

Therefore, (3.1.46) and (3.1.47) imply that the sequence ¹wn,k º1 nD1 is uniformly integrable in L1 ./ and the sequence ¹h ı wn,k º1 is uniformly integrable in nD1 L1.; gr ˇ C1 /. By a standard argument, a subsequence of ¹wn,k º1 converges pointnD1 wise a.e. in  and the limit of this sequence is a solution of (3.1.1) with with boundary trace kıy , i.e., uk,y . Since this solution is unique we conclude that the entire sequence ¹wn,k º1  nD1 converges. Finally, by (3.1.45), we conclude that uk,y  u. Remark 3.1.15. In addition to the assumptions of the theorem, suppose that uk,y,g ! 1

in  8y 2 @.

(3.1.48)

Then every positive solution u of (3.1.1) is moderate because the last theorem implies that S.u/ D ;.

3.1.3 Unbounded domains Most of the results of this section, with minor modifications, remain valid for unbounded domains, uniformly of class C 2. In some cases an additional condition is needed. We use here the notation introduced in Section 2.5. For unbounded domains Definition 3.1.1 should be modified as follows: Definition 3.1.18. A solution u of (3.1.1) is moderate if, for every C 2 bounded domain D   there exists a positive harmonic function U D such that juj  U D in D. With this modification, Theorem 3.1.2 remains valid provided that L1 .; / is replaced by .L1 /0 ./. The same remark applies to Proposition 3.1.6. Proposition 3.1.4, being of a local nature, applies to unbounded domains without any modification. Theorem 3.1.8 remains valid if in part (i) we assume that supp h is bounded. Theorem 3.1.12, the definitions of singular points and rough traces and the remaining results in Section 3.1 remain valid as stated.

3.2

Maximal solutions

In this section  denotes a bounded or unbounded domain, uniformly of class C 2. Throughout the section we shall assume that g 2 G0 and that it satisfies the local

79

Section 3.2 Maximal solutions

Keller–Osserman condition and the global barrier condition, see Definitions 3.1.14 and 3.1.18. The local Keller–Osserman condition guarantees that (3.1.1) possesses a maximal solution in . Since  is of class C 2 , the maximal solution blows up at every point of the boundary, i.e., it is a large solution. Notation. If u, v are two positive solutions of (3.1.1) then the smallest solution dominating the subsolution max.u, v/ is denoted by u _ v. The largest solution dominated by the supersolution min.u, v/ is denoted by u ^ v. For the existence of these solutions apply Theorem 2.2.7 using the fact that there exists a maximal solution in . We start with an elementary lemma (see [61] or [84, Proposition 3.3]) which plays an important role in the sequel. Lemma 3.2.1. Let V be a family of positive solutions of (3.1.1). (I) If V satisfies u, v 2 V H) u _ v 2 V

(3.2.1)

then U :D sup V is a solution of (3.1.1) and U is the limit of a non-decreasing sequence of solutions belonging to V. (II) If V satisfies

u, v 2 V H) u ^ v 2 V

(3.2.2)

then U0 :D inf V is a solution of (3.1.1) and U0 is the limit of a non-increasing sequence of solutions belonging to V. Proof. (I)

First we prove the assertion under the additional assumption:

If ¹un º is a non-decreasing sequence in V then lim un 2 V. As a consequence of the local Keller–Osserman condition, the family of all positive solutions of (3.1.1) is uniformly bounded in every compact subset of . Pick x0 2  and a sequence ¹uk º  V such that uk .x0/ ! U.x0/. For each k we define wk D u1 _    _ uk . Then wk 2 V and ¹wk º increases. By assumption w D lim wk 2 V. If v 2 V then v _ w 2 V and consequently .v _ w/.x0/  w.x0/. On the other hand w  v _ w. Therefore .v _ w/.x0/ D w.x0 / and, by the maximum principle, v _ w D w. It follows that v  w; hence w D U . Now, if V is not closed with respect to monotone non-decreasing convergence, let VQ be the set of all limits of non-decreasing sequences in V. It is easy to verify that VQ satisfies (3.2.1) and that it is closed with respect to monotone non-decreasing convergence. In addition sup VQ D sup V. Therefore the assertion of the lemma follows from the first part of the proof. (II) Except for obvious modifications, the proof is the same as in part I. 

80

Chapter 3 The boundary trace and associated boundary value problems

Definition 3.2.2. Let F be a compact subset of @ and denote by VF the family of all positive solutions u of (3.1.1) such that N nF/ u 2 C.

and u D 0 on @ n F .

We say that U is the maximal solution relative to F if U 2 VF and U dominates every solution in VF . Lemma 3.2.3. Let F be a compact subset of @. Then UF :D sup VF belongs to VF . Therefore it is the maximal solution relative to F . Remark 3.1. It is possible that VF contains only the trivial solution. In this case the conclusion is trivial. Proof. Let z 2 @ n F and let s D 12 min.r .z/, dist .z, F //. Then the function ws,z defined in Definition 3.1.10 dominates, in Bs .z/ \ , every solution belonging to VF . Therefore, if u, v 2 VF then u _ v 2 VF and if u is the limit of a sequence in VF then u 2 VF . Consequently, by Lemma 3.2.1, UF 2 VF .  Remark 3.2.3. Clearly, S.UF / D: F 0  F . However F 0 need not be equal to F . For instance, suppose that g is supercritical – so that the equation has no solution with point singularity – and that F is a singleton. Then UF D 0. We denote kg .F / D S.UF /.

(3.2.3)

Lemma 3.2.4. The mapping F 7! UF is monotone: if F1 , F2 are compact subsets of @ then (3.2.4) F1  F2 H) UF1  UF2 If F is a compact subset of @ and ¹Nk º is a decreasing sequence of relatively open neighborhoods of F such that NN kC1  Nk and \Nk D F then UNN k ! UF

(3.2.5)

uniformly in compact subsets of . In addition to the basic assumptions on g detailed at the beginning of this section, assume that, for each x 2 , g.x, / is convex. If F1 , F2 are two compact subsets of @ then UF1[F2  UF1 C UF2 . (3.2.6) Proof. The first statement is an immediate consequence of the definition of maximal solution. Next we verify (3.2.5). By (3.2.4) the sequence ¹UNN k º decreases and therefore it converges to a solution U . Clearly U has trace zero outside F so that U  UF . On the other hand, for every k, UNN k  UF . Hence U D UF .

81

Section 3.3 The boundary value problem with rough trace

We turn to the verification of (3.2.6). Let Q1, Q2 be open neighborhoods of F1 , F2 respectively. Put U :D UF1 [F2 . For ˇ 2 .0, ˇ0 / put Qi ,ˇ D ¹x 2  : .x/ D ˇ, .x/ 2 Qi ,

hi ,ˇ D U 1Qi,ˇ ,

i D 1, 2.

Let vi ,ˇ be the solution of (3.1.1) in Dˇ such that vi ,ˇ D hi ,ˇ on †ˇ . Let vˇ be the solution in Dˇ with boundary data hˇ D U  max.h1,ˇ , h2,ˇ /. Then U  wˇ :D vˇ C v1,ˇ C v2,ˇ

on †ˇ .

Since g.x, / is convex, non-negative and vanishes at zero, g.x, a/ C g.x, b/  g.x, a C b/ 8a, b > 0.

(3.2.7)

Therefore the sum of two positive solutions of (3.1.1) is a supersolution. Thus wˇ is a supersolution of (3.1.1) in Dˇ and U  wˇ on †ˇ . Consequently U  wˇ

in Dˇ .

Let ¹ˇn º be a sequence decreasing to zero such that ¹vˇn º, ¹v1,ˇn º, ¹v2,ˇn º converge. Since hˇ ! 0 (uniformly) vˇn ! 0. Put vi :D lim vi ,ˇn . Then vi is a solution of (3.1.1) in  and (because of the global barrier condition) vi D 0 on @ n QN i . Therefore U  v1 C v2  UQN 1 C UQN 2 in . 

In view of (3.2.5), this implies (3.2.6).

3.3 The boundary value problem with rough trace The next theorem provides necessary and sufficient conditions for the existence and uniqueness of solutions of the generalized boundary value problem: u C g ı u D 0, Tr u D , N

u  0 in ,

(3.3.1)

where N 2 Breg . Let ., F / be the characteristic representation of N (see Definition 3.1.13); thus F  @ is a closed set and  is a (non-negative) Radon measure on @ n F . Theorem 3.3.1. Let  be a bounded domain of class C 2 and let g 2 G0. Assume that g.x, / is convex for every x 2  and that g satisfies the local Keller–Osserman condition and the barrier condition (see Definition 3.1.10). E XISTENCE . The following set of conditions is necessary and sufficient for existence of a solution u of (3.3.1):

82

Chapter 3 The boundary trace and associated boundary value problems

(i) For every compact set E  @ n F , the measure 1E is g-good. (ii) F nkg .F /  S.V /. Here kg .F / is defined as in (3.2.3) and V D V ,g is defined as in (3.1.26). If (i) and (ii) hold then any solution u satisfies V  u  V C UF .

(3.3.2)

Furthermore if F is a removable set then (3.3.1) possesses exactly one solution. U NIQUENESS . Assume that UF is the unique solution with trace .0, kg .F //.

(3.3.3)

(a) If u is a positive solution of (3.1.1) such that S.u/ D F then max.V , UF /  u  V C UF ,

(3.3.4)

where  denotes the regular part of the boundary trace of u. (Note that, by the first part of the theorem, ., F / must satisfy (i) and (ii).) (b) Equation (3.1.1) possesses at most one solution satisfying (3.3.4). Thus if condition (3.3.3) holds for a given set F then (3.3.1) possesses at most one solution for every positive Radon measure  on @ n F . M ONOTONICITY. Let u1 , u2 be two positive solutions of (3.1.1) with boundary traces .1 , F1 / and .2 , F2 / respectively. Suppose that F1  F2 ,

1  2 1@nF2 D: 20 in @ n F2 .

(3.3.5)

If (3.3.3) holds for F D Fi , i D 1, 2, then u1  u2 . Proof. First assume that there exists a solution u of (3.3.1). Since V  u the function w :D u  V is a subsolution of (3.1.1). This is a consequence of (3.2.7). By Theorem 2.2.7, as w is dominated by the maximal solution in , there exists a solution wN of (3.1.1) which is the smallest solution dominating w. N such that h D 0 on a neighborhood of S.u/, For every h 2 C./ Z lim whdS D 0. ˇ !0 †ˇ

Following the construction in Theorem 2.2.7, it is easy to check that wN has the same property. Thus Tr wN vanishes on R.u/.

Section 3.3 The boundary value problem with rough trace

83

The definitions of V and wN imply N  u  V C w. N max.V , w/

(3.3.6)

Therefore S.w/ N [ S.V / D S.u/. In addition, as wN has trace zero in @ n F , it follows that wN  UF and consequently S.w/ N  kg .F / where UF is the maximal solution relative to F . These observations imply that u satisfies condition (ii). Inequality (3.3.2) follows from (3.3.6) and this inequality implies that if F is a removable set then (3.3.1) possesses exactly one solution. Now we assume that conditions (i) and (ii) hold and prove existence. The function V is well defined and V CUF is a supersolution of (3.1.1) while max.UF , V is a subsolution. Both have boundary trace ., F /. Therefore the largest solution dominated by V C UF has the same boundary trace, i.e. solves (3.3.1). By the definition of trace, V  u for every solution u of (3.3.1). Furthermore uV is a subsolution which vanishes on @ n F . Therefore u  V  UF . Next assume that (3.3.3) holds. We must show that, if there exists a positive solution u of (3.1.1) with boundary trace ., F / then (a) and (b) hold. The existence of such a solution implies that conditions (i) and (ii) hold and UF  u. Consequently (3.3.2) implies (3.3.4) so that (a) holds. We turn to the proof of (b). Let u be the smallest solution dominating the subsolution max.V , UF / and let v be the largest solution dominated by V C UF . Then, (3.3.4) implies that u  v and v  u  min.V , UF /.

(3.3.7)

Let ¹Nk º be a decreasing sequence of open sets converging to F such that NkC1 b Nk . Assuming for a moment that  is a finite measure, the trace of V on Nk is 1Nk and it tends to zero as k ! 1. Therefore, in this case, min.V , UF /  V 1Nk ! 0 and hence u D v. This implies uniqueness in the case where  is a finite measure. In the general case we argue as follows. Let k :D 1@nNk . This measure is finite and consequently there exists a unique solution vk with boundary trace .k , NN k /. Taking a subsequence we may assume that ¹vk º converges to a solution v 0 . By (3.3.4), max.V k , UNN k /  vk  V k C UNN k . Here we need the following: Assertion 1. Suppose that .i , Fi /, i D 1, 2 satisfy (3.3.5). If w is a positive solution of (3.1.1) then (3.3.8) w  V1 C UF1 H) w  V2 C UF2 .

84

Chapter 3 The boundary trace and associated boundary value problems

By Lemma 3.2.4, V1 C UF1  V2 C V11F2 nF1 C UF2  V2 C 2UF2 . Therefore w  V1 C UF1 H) .w  V2 /C  2UF2 . Thus .w  V2 /C is a subsolution, 2UF2 is a supersolution and UF2 is the largest solution dominated by 2UF2 . Therefore .w  V2 /C  UF2 . Since v  V C UF , Assertion 1 implies that v  V k C UNN k . But vk is the largest solution dominated by the right-hand side of this inequality. Therefore v  vk and consequently v  v 0 . By (3.2.5) UNN k # UF and by definition V k " V . Therefore max.V , UF /  v 0  V C UF . Since v is the largest solution dominated by V CUF and v  v 0 it follows that v D v 0 . Let uk be the unique solution with boundary trace .k , kg .F //. By (3.3.4), max.V k , Ukg .F / /  uk  V k C Ukg .F / . The definition of u implies that uk  u. Since ¹V k º increases it follows that ¹uk º increases. Therefore u0 :D lim uk  u and (recall that Ukg .F / D UF ) max.V , UF /  u0  V C UF . As u0  u and u is the smallest solution dominating max.V , UF / it follows that u0 D u. Let uk,j be the unique solution with boundary trace .k , NNj /, j  k. By (3.3.4), max.V k , UNNj /  uk,j  V k C UNNj . When j " 1, uk,j # u0k where u0k is a solution of (3.1.1) with trace .k , kg .F //. Because of the uniqueness of such solutions, u0k D uk . Therefore one can obtain an increasing sequence ¹jk º such that k C 1 < jk and wk :D uk,jk ! u. Clearly wk < vk . We show that lim vk  wk D 0.

k!1

85

Section 3.3 The boundary value problem with rough trace

By Lemma 3.2.4 and (3.3.4), vk  V k C UNN k  V k C UNN k nNj C UNNj , k

k

wk  max.V k , UNNj /. k

Hence, vk  wk  min.V k C UNN k nNj , UNN k nNj C UNNj /  zk C k k

k

k

where zk :D min.V k , UNN k nNj C UNNj / D 0 on @ k

k

and k :D UNN k nNj ! 0 as k ! 1. k

Thus .vk  wk  k /C is a subsolution of (3.1.1) dominated by zk which vanishes on the boundary. It follows that vk  w k   k . Since u  v we conclude that v  u D lim.vk  wk / D 0. Finally we establish monotonicity. Let vi be the unique solution of (3.1.1) with boundary trace .i , Fi /, (i=1,2). Then vi is the largest solution dominated by V i CUFi (i=1,2). Therefore, by Assertion 1, v1  V 2 C UF2 . As v2 is the largest solution dominated by the right-hand side it follows that v1  v2 .  Definition 3.3.2. Let  be a (possibly unbounded) domain, uniformly of class C 2 and let g 2 G0 ./. We say that g satisfies the strong local Keller–Osserman condition if it satisfies the local Keller–Osserman condition and the following additional condition: For every R > 0 let uR denote the maximal solution of (3.1.1) in  \ BR .0/ which vanishes on @ \ BR .0/. Then lim uR D 0

R!1

N uniformly in compact subsets of . The above condition is satisfied by any g 2 G0 ./ for which the global Keller– Osserman condition holds. However, there are interesting examples of nonlinearities g 2 G0./ which do not satisfy the global condition but nevertheless satisfy the strong local condition. One such example, discussed in detail in Section 3.4, is given by g.x, t / D .x/˛ jt jq sign t where ˛ > 0 and q > 1. Corollary 3.3.3. Let  be a (possibly unbounded) domain, uniformly of class C 2 and let g 2 G0 ./. Assume that, in addition to the conditions of the theorem, g satisfies the strong local Keller–Osserman condition. Then the assertions of the theorem remain valid.

86

Chapter 3 The boundary trace and associated boundary value problems

Proof. If Q  @ is a relatively open set, define: Q .E/ D .E/

8E  QN n F , E Borel

N [ S.V Q /. FQ D kg .F \ Q/ Let ¹n º be an increasing sequence of bounded domains of class C 2 such that (i) n " . (ii) There exists a sequence ¹Rn º strictly increasing to infinity such that  \ BN Rn .0/  n ,

N n1  BRn .0/. 

(iii) There exists a relatively open set Qn  @ such that .@n1 \ @/  Qn  QN n  BRn . Let

´ Qn n D 0

in QN n in @n n QN n .

and Fn D FQn . Let N n 2 Breg .@n / be the measure represented by .n , Fn /. n " ,

Fn " F .

If N satisfies conditions (i) and (ii) then n " ,

Fn " F

and N n satisfies these conditions relative to n . Therefore problem u C g ı u D 0, u  0 Tr u D N n

in n ,

(3.3.9)

has a solution and every solution satisfies the inequality corresponding to (3.3.2). Consequently, the maximal solution uN n is the largest solution in n dominated by V n C UFn . Furthermore uN n , extended by zero to nC1 , is a subsolution of the corresponding problem in nC1 . Therefore ¹uN n º is increasing and the limit uN is a solution of (3.3.1) and satisfies (3.3.10) uN  V C UF . The necessity of conditions (i) and (ii) for existence is verified as in the proof of the theorem. Assertion. Suppose that conditions (i) and (ii) and (3.3.3) hold. If v is a positive solution of (3.3.9) then UF \QN n1  v. (3.3.11)

Section 3.4 A problem with fading absorption

87

Note that v C UF nQn is a supersolution in  whose singular boundary set contains kg .F /. Therefore (3.3.3) implies that UF  v C UF nQn . But UF  UF nQn is a subsolution whose singular boundary set contains F \ QN n1 . Since v is a solution which dominates this subsolution we obtain (3.3.11). Consequently, under the assumptions of this assertion, the minimal solution of (3.3.9), say un , satisfies max.V n , UF \QN n1 /  un . The sequence ¹un º increases and u D lim un is a solution of (3.3.1) in . In view of (3.3.3), UF  u. These facts and (3.3.10) imply max.UF , V /  u  uN  UF C V . It follows that uN  u  min.UF , V / which implies that uN  u C UF and therefore uN  u. Hence, uN D u.

(3.3.12)

Let wn be the maximal solution of (3.1.1) in n vanishing on Qn . In view of (3.2.7), uN n C wn is a supersolution of (3.1.1) in n . Therefore, if u is a positive solution of (3.3.1) then un  u  uN n C wn in n . Since g satisfies the strong local Keller–Osserman condition, wn ! 0 as n ! 1. Therefore u  u  u. N In view of (3.3.12) we conclude that, if (i), (ii) and (3.3.3) hold, problem (3.3.1) possesses a unique solution. We have already observed that (i) and (ii) are necessary and sufficient for existence and, evidently, (3.3.3) is necessary for uniqueness. The monotonicity is verified as in the proof of the theorem. 

3.4 A problem with fading absorption We apply the results of the previous sections to a study of boundary value problems for equation u C g ı u D 0 in  (3.4.1) where g.x, t / D .x/˛ jt jq sign t

(3.4.2)

88

Chapter 3 The boundary trace and associated boundary value problems

˛ > 2, q > 1 and  is a uniformly C 2 domain. We observe that under these conditions g 2 G0. Therefore the boundary value problem u C  ˛ jujq sign u D 0 uD

in  on @

(3.4.3)

possesses a solution for L1 boundary data and in fact all the results presented in Sections 2.1, 2.2 and 2.4 apply. Furthermore, the local Keller–Osserman condition is satisfied; hence the set of solutions of (3.4.1) is uniformly bounded in every compact subset of . The case ˛ D 0 is of particular interest because of its significance in probability (when 1 < q  2), and other applications. In this case the global Keller–Osserman condition holds and therefore at every point y 2 @ there exists a barrier, and for every compact set F  @ there exists a maximal solution (see Definitions 3.2.2 and 3.1.10). This is also true for 2 < ˛ < 0 because in this case .x/˛ jt jq  jt jq

for .x/ < 1.

However in the case ˛ > 0, the nonlinearity fades at the boundary and therefore the global Keller–Osserman condition is not satisfied. Therefore it is not at all obvious that there exist barriers and maximal solutions in the sense of Definition 3.2.2. Indeed it is known that, if the coefficient in the absorption term fades at the boundary at the rate of exp .1=/ then any solution which blows up at one point of @ blows up everywhere on @. Nevertheless, we shall see that, when g is given by (3.4.2), barriers do exist.

3.4.1 The similarity transformation and an extension of the Keller–Osserman estimate A basic tool in our presentation is a similarity transformation associated with (3.4.1). q,˛ This transformation, denoted by Ta , a > 0 is given by 2C˛

Taq,˛ u.x/ D a q1 u.ax/.

(3.4.4)

If u is a solution of (3.4.1) in a domain  then Taq,˛ u is a solution of this equation q,˛ in a1 . If  D a1  and u D Ta u for every a > 0 we say that u is a self-similar solution. Although g does not satisfy the global Keller–Osserman condition, the solutions of (3.4.1) satisfy a global Keller–Osserman type estimate. The next lemma establishes such an estimate, using the similarity transformation (3.4.4). Lemma 3.4.1. There exists a constant C depending only on N , q, ˛ (q > 1, ˛ > 2) and the C 2 characteristic of  such that every solution u of (3.4.1) satisfies 2C˛

ju.x/j  C.x/ q1

8x 2  : .x/ < ˇ0 .

(3.4.5)

89

Section 3.4 A problem with fading absorption

Proof. Since juj is a subsolution, it is sufficient to prove the lemma for u > 0. Pick y 2  and put R D .y/=2. As usual, BR .y/ denotes the open ball of radius R centered at y. If x 2 BR .y/ then R < .x/ < 3R. Therefore, g.x, t / D  ˛ .x/t > c0 .˛/R˛ t where ´ 1 if ˛  0, c0 .˛/ D ˛ if ˛ 2 .2, 0/. 3 Therefore, if vR is the maximal solution of v C c0 R˛ v q D 0 in BR .0/ then u.x/  vR .x  y/ 8x 2 BR .y/. By the results of Subsection 4.1.2: vR .0/  c.N , q/.c0 .˛/R2C˛ / q1 . 1

Therefore

2C˛

u.y/  C.N , q, ˛/R q1 .

This inequality implies (3.4.5).



Remark 3.2. It should be noted that the estimate (3.4.5) by itself does not lead to the existence of barriers. The reason is that, contrary to the case where g satisfies the global Keller–Osserman condition, this estimate is confined to .

3.4.2 Barriers and maximal solutions The next result establishes the existence of barriers for g as in (3.4.2). Proposition 3.4.2. Let  be a domain uniformly of class C 2. Let z 2 @, 0 < r < ˇ0 =2 and let g be as in (3.4.2) with ˛ > 0, q > 1. Then there exists a positive solution W D Wz,r of (3.4.1) in Br .z/ \  such that W 2 C.Br .z/ \ /, W D 0 on @ \ Br .z/, limx!y W .x/ D 1 8y 2 @Br .z/ \ .

(3.4.6)

N \ Bˇ =2 .z// is Furthermore, if r D 3ˇ0 =4, the norm of the solution Wz,r in C 2. 0 2 bounded by a constant depending only on N , ˛, q and the C characteristic of  but not on z. Proof. Let f be the function defined in BN r .z/ by f .x/ D r 2  jx  zj2 . Put w D f ˇ   where ˇ < 0 and 2 .0, 1/. Clearly w satisfies (3.4.6). We shall show that ˇ, can be chosen in such a way that w is a supersolution of (3.4.1) in a one-sided neighborhood of BN r .z/ \ @.

90

Chapter 3 The boundary trace and associated boundary value problems

From the definition of w we obtain,   w D ˇ.ˇ  1/f ˇ 2 jrf j2 C ˇf ˇ 1f   C 2ˇ f

ˇ 1 1



ˇ 2

rf  r C .  1/f 

(3.4.7) ˇ 1

C f 

.

Hence, 

 ˇ.ˇ  1/ jrf j2 C ˇf f  2 C 2ˇ frf  r  C .  1/f 2 C f 2 f ˇ 2  2  g ı w.

w  g ı w 

(3.4.8)

Given  > 1 and 0 < < r=2 put D. / D ¹x 2 Br .z/ \  : .x/ < º DC ., / D ¹x 2 D. / : f .x/ > º D ., / D ¹x 2 D. / : f .x/  º.

(3.4.9)

By (3.4.8) the following inequality holds in DC., /: w  g ı w  .M= C .  1//f ˇ  2,

(3.4.10)

N norm of the distance function where M depends on ˇ, , r and the C 2.Br .z/ \ / , but not on  and . Choosing  > 1 sufficiently large, e.g.  D 2M= .1  /, we obtain w  g ı w  0 in DC ., /. (3.4.11) Next choose and ˇ so that

2 .0, 1/,

ˇ<

2C˛  . q1

(3.4.12)

Note that this choice determines the constant M in (3.4.10). In D ., / (3.4.8) yields the following inequality w  g ı w  M 0 f ˇ 2     ˛Cq f qˇ , N norm of , where M 0 is a constant depending on , ˇ, , r and the C 2.Br .z/ \ / but not on . Hence,   (3.4.13) w  g ı w  f ˇ 2  2 M 0  2   ˛C.q1/C2f ˇ .q1/C2 . Since, by (3.4.12), ˇ.q  1/ C 2 < 0 it follows that, in D ., /,   w  g ı w f ˇ 2  2 M 0  2   ˇ.q1/C2  ˛C.Cˇ /.q1/C4   f ˇ 2   M 0   ˇ.q1/C2  ˛C.Cˇ /.q1/C2 .

(3.4.14)

Section 3.4 A problem with fading absorption

91

In addition, by (3.4.12), ˛ C . C ˇ/.q  1/ C 2 < 0. Therefore there exists .r / > 0, N norm of , such that if 2 depending only on , ˇ, , r and the C 2 .Br .z/ \ / .0, .r //, (3.4.15) w  g ı w  0 in D ., /. The choice of ˇ, depends only on q and ˛ while the choice of  and M 0 depends N norm of . This norm is bounded by a constant in addition on the C 2 .Br .z/ \ / 2 determined by the C characteristic of the domain and r and is independent of z. Therefore .r / depends only on r , ˛, q and the C 2 characteristic of  but not on z. In conclusion, if 2 .0, .r //, w  g ı w  0, w > 0 in D. /, w D 0 on @Br .z/ \ @, limx!y w.x/ D 1 8y 2 @Br .z/ \  : .y/ < .

(3.4.16)

Since g satisfies the local Keller–Osserman condition, there exists a solution v of the problem v  g ı v  0, in D. /, (3.4.17) v D 0 on ¹x 2 @D. / : .x/ < º, limx!y w.x/ D 1 8y 2 @D. / : .y/ D . The function wN :D v C w is a positive supersolution in D. / which vanishes on @ \ Br .z/ and blows up at @D. / \ . Let wk be a solution of the problem wk  g ı wk D 0, in Br .z/ \ , wk D 0 on Br .z/ \ @, wk D k on @Br .z/ \ .

(3.4.18)

N Consequently Then ¹wk º is an increasing sequence dominated by w. W :D lim wk k!1

is a positive solution of (3.4.1) in Br .z/ satisfying (3.4.6).



Remark 3.3. Note that this proposition also establishes the fact that the nonlinearity g given by (3.4.2) satisfies the global barrier condition (see Definition 3.1.10). Since g satisfies the local Keller–Osserman condition, Lemma 3.2.3 and Proposition 3.4.2 imply: Corollary 3.4.3. Let F  @ be a compact set. If u is a solution of (3.4.1) vanishing on @ n F then there exists solution uN vanishing on @ n F such that juj  u. N Furthermore there exists a solution UF of (3.4.1) such that UF vanishes on @ n F and UF dominates every solution with this property.

92

Chapter 3 The boundary trace and associated boundary value problems

Proof. If u is a solution of (3.4.1) then juj is a subsolution. In view of Proposition 3.4.4 the smallest solution dominating juj vanishes on @ n F . Let UF denote the family of positive solutions of (3.4.1) vanishing on @ n F . If u, v 2 UF then max.u, v/ is a subsolution and, as before, the smallest solution dominating it belongs to UF . Therefore, by Lemma 3.2.3, UF :D sup UF is a solution of (3.4.1) and, in view of Proposition 3.4.2, UF vanishes on @ n F .  Using Corollary 3.4.3 we obtain the following significant improvement of inequality (3.4.5). Proposition 3.4.4. Let  be a domain (not necessarily bounded) uniformly of class C 2 and let F be a compact subset of the boundary. Then, for q > 1 and ˛ > 2, there exists a constant C depending only on N , q, ˛ and the C 2 characteristic of  such that, for every solution u of (3.4.1) vanishing on @ n F 2C˛

ju.x/j  C.x/dist .x, F / q1 1

(3.4.19)

for every x 2  such that dist .x, F /  .1 C ˇ0 /1 . If  is bounded the inequality holds for every x 2 . Proof. In view of Corollary 3.4.3 we may assume that u is a positive solution. Let P 2 @ n F and put 1 dF .P / D dist .P , F /. 2 P Let  be a local set of coordinates at P with the properties described in Definition 1.1.1. Denote P D

1 , dF .P /

D P D P \ Bˇ0 =2 .P /.

If u is a solution of (3.4.1), denote by uP the function 2C˛

uP ./ D jdF .P /j q1 u.dF .P // Then

8 2 P .

uP ./ C dist ., @P /˛ .uP .//q D 0 8 2 P .

Let

 3ˇ0 1  min 1, . 4 dF .P / We assume that dF .P /  1 so that r D 3ˇ0 =4. Then the solution WP ,r mentioned in Proposition 3.4.2 satisfies uP < WP ,r in D P . r D r .P / D

93

Section 3.4 A problem with fading absorption

Thus uP is bounded in D P by a constant C depending only on N , q, ˛ and the C 2 characteristic of P . As dF .P /  1 a C 2 characteristic of  is (up to a constant independent of P ) also a C 2 characteristic of P . Therefore the constant C can be taken to be independent of P . If ˛  0 then  ˛ .uP /q1 is bounded in Bˇ0=2 .P / \ P . Therefore, as uP D 0 on @ \ Br .P /, Hopf’s lemma implies that uP ./  C dist ., @P /

8 2 D P .

Hence, 2C˛

u./  C dist ., @/dF .P / q1 1

8 2 Bd

ˇ0 F .P / 2

.P / \ .

(3.4.20)

Next we consider the case 2 < ˛ < 0. Let v P denote the solution of v C v q D 0 in D P ,

v D uP

on @D P .

Since ˛ < 0, uP is a subsolution of this boundary value problem. Therefore uP < v P in D P . However, by the previous argument, v P ./  C dist ., @P / 8 2 D P . Thus (3.4.20) holds for ˛ > 2. Let x 2 ˇ0 and assume that .x/ D dist .x, @/  ˇ0 dist .x, F /,

dist .x, F / <

1 . 1 C ˇ0

Let P be the unique point on @ for which dist .x, P / D .x/. Then dF .P /  .x/ C dist .x, F /  .1 C ˇ0 /dist .x, F / < 1 and

2C˛

u.x/  C.x/..1 C ˇ0 /dist .x, F // q1 1 .

On the other hand, if .x/  ˇ0 dist .x, F / then, by Lemma 3.4.1, 2C˛

2C˛

u.x/  C.x/ q1  C.x/.ˇ0dist .x, F // q1 1 . Thus (3.4.19) holds for every x 2 ˇ0 such that dist .x, F / < .1 C ˇ0 /1 . If  is bounded then, by the maximum principle, u is bounded in the set ¹x 2  : dist .x, F /  .1 C ˇ0 /1 º. Therefore, by Hopf’s lemma, (3.4.19) holds in this set and consequently, for every x 2 . 

94

Chapter 3 The boundary trace and associated boundary value problems

In the remaining part of this section we employ the results just established in order to prove two main results. The first determines the critical value of the exponent in (3.4.1). The second presents a complete classification of positive solutions in the subcritical case in terms of the rough trace. In addition we obtain sharp estimates of positive solutions as x tends to the critical set of the solution on the boundary. In both cases  is a domain in RN (possibly unbounded) uniformly of class C 2.

3.4.3 The critical exponent Theorem 3.4.5. Let

N C1C˛ . (3.4.21) N 1 (i) If 1 < q < qc then the boundary value problem (3.4.3) possesses a (unique) solution for every  2 M.@/. If  is unbounded this statement remains valid if  is a Borel set function on @, not necessarily positive, such that, for every compact set E  @, 1E is a finite measure. qc D qc .˛/ :D

(ii) Let y 2 @. If q  qc then isolated point singularities are removable, i.e., if v is a solution of (3.4.1) vanishing on @ n ¹yº then v 0 in . Proof. (i) First assume that  is bounded. Let y 2 @ and let P D P denote the Poisson kernel for  in . A simple computation shows that there exists a constant c1 depending only on N , q, ˛ and  such that Z P .x, y/q .x/˛C1 dx < c1 if 1 < q < qc . 

Let  be a finite Borel measure on @ and let v be the harmonic function with boundary trace jj: Z v.x/ D Then

Z

Z 

v q  ˛C1dx  c2

@

P .x, y/ d jj.y/,

Z @



8x 2 .

 P .x, y/q .x/˛C1dx d jj.y/  c3 < 1.

Therefore v is a (weak) supersolution of problem (3.4.3) while v is a (weak) subsolution. Hence, by Theorem 2.2.4, this problem has a solution. Since  is a finite measure uniqueness holds. In the case of an unbounded domain, assume at first that  is a positive Radon measure. Let ¹n º be a sequence of subdomains as in Corollary 3.3.3 and let n be defined as in the proof of that corollary. As proved above, there exists a unique solution un in n with boundary data n . The sequence ¹un º increases and its limit u is a solution of (3.4.3).

95

Section 3.4 A problem with fading absorption

Let wn be defined as in the proof of Corollary 3.3.3. Then Lemma 3.4.1 implies that lim wn D 0. If uQ is a solution of (3.4.3) then un  wn  uQ  un C wn

in n .

We conclude that uQ D u. Now let  be a Borel set function on @, not necessarily positive, such that, for every compact set E  @, 1E is a finite measure. Let un be the unique solution in n with boundary data n . Since g satisfies the local Keller–Osserman condition, we can extract a subsequence (still denoted by ¹un º) which converges to a solution u of (3.4.3). If u, v are two solutions of (3.4.3) then u  v has zero M-boundary trace. Therefore wn  u  v  wn in n , n D 1, 2, : : : . Hence u D v. (ii) Since jvj is a subsolution, it is sufficient to prove the lemma in the case that v  0. We may of course assume that y D 0. We show that, if  is bounded, v 2 Lq .;  ˛C1 / \ L1./.

(3.4.22)

Let  be a function in C 2 .R/ such that 0    1,

.t / D 0 for t < 1,

.t / D 1 for t > 2.

Further let be the solution of the problem  D 1 in ,

D 0 on @.

(3.4.23)

Given > 0 put  .x/ D .jxj = / .x/ for x 2 . Since  is a C 2 test function which vanishes on @ and in a neighborhood of the origin we have Z .v C  ˛ v q  /dx D 0. (3.4.24) 

If E D ¹x 2  : < jxj < 2 º then Z Z Z v dx  C1 v. 2 C 1 /dx  v.x/.x= / dx  E  Z Z  C2 v 1 dx  v.x/.x= / dx, E

(3.4.25)



where C1, C2 are constants independent of . Here we used the fact that D O. / in E . From (3.4.24) and (3.4.25) we obtain, Z Z .v.x/.x= / C  ˛ v q  /dx  C2 v 1dx. (3.4.26) 

E

96

Chapter 3 The boundary trace and associated boundary value problems

By Lemma 3.4.1 2C˛

v.x/  c.x/ jxj q1 1 . Since q  qc H)

2C˛ N 1 q1

it follows that v.x/  c N C1

in E .

Consequently there exists a constant C3 independent of such that Z v 1 dx  C3 , 8 > 0.

(3.4.27)

E

By (3.4.26) and (3.4.27) Z .v.x/ C  ˛ v q /.x= /dx  C4 , 

8 > 0,

where C4 is independent of . Note that D O./ and .x= / D 1 for jxj > 2 . Therefore letting ! 0, the previous inequality and Fatou’s lemma imply (3.4.22). Now suppose that v > 0 and let V denote the maximal solution of (3.4.1) vanishing on @ n ¹0º. Since V satisfies (3.4.22), its boundary trace must be a bounded measure concentrated at 0, say ı0 . It follows that 2V is a supersolution of (3.4.1) with Mboundary trace 2 ı0 and the largest solution dominated by 2V say V 0 has the same boundary trace. This implies V < V 0 , which is impossible as V was the maximal solution vanishing on @ n ¹0º. This proves the lemma in the case that  is bounded. If  is unbounded we pick a sequence of domains ¹n º and of solutions ¹wn º as in the proof of Corollary 3.3.3. We observe that v  wn in n . Thus v D 0. 

3.4.4 The very singular solution In the remainder of this section we study equation (3.4.1) in the subcritical case, i.e., under the assumption 1 < q < qc . (3.4.28) By Theorem 3.4.5, in this case problem (3.4.3) possesses a unique solution for every  2 M.@/. Notation and Terminology. If  is a finite Borel measure on @ we denote by u./ the unique solution of (3.4.3). If y 2 @ and k 2 R we denote by uk,y the unique solution of (3.4.3) with  D kıy . Finally we denote u1,y D lim uk,y . k!1

(3.4.29)

97

Section 3.4 A problem with fading absorption

Let Uy ./ denote the space of positive solutions of (3.4.1) such that N n ¹yº/, u 2 C.

u D 0 on @ n ¹yº.

(3.4.30)

Put U1,y D sup¹u 2 Uy ./º.

(3.4.31)

By Corollary 3.4.3, U1,y is a solution of (3.4.1) and satisfies (3.4.30). Therefore it is the maximal element of Uy ./. A positive solution u of (3.4.1) satisfying (3.4.30) and the singularity condition Z uq  ˛C1 dx D 1 (3.4.32) \Br .y/

is called a very singular solution at y. In other words, a positive solution v is a very singular solution at y if S.v/ D ¹yº and v D 0 on @ n ¹yº. Equivalently, v is a positive solution with boundary trace 1ıy . Theorem 3.4.6. Let  be a (possibly unbounded) domain, uniformly of class C 2 and let y 2 @. Assume that 1 < q < qc and 0  ˛. (i) Equation (3.4.1) possesses a unique (positive) very singular solution at y. This solution will be denoted by Uy . (ii) There exists a positive constant c, depending only on N , q the C 2 characteristic of  and the modulus of continuity associated with the C 2 representation of @ (see Definition 2.5.1) such that 2C˛ 2C˛ 1 jx  yj q1 1 .x/  Uy .x/  cjx  yj q1 1 .x/ c

(3.4.33)

for every x 2  \ Br0 =4 .y/. (Recall that r0 is the first component of the C 2 characteristic of , see Definition 1.1.1.) If  is bounded then (3.4.33) holds everywhere in  with a constant c which may depend also on diam . (iii) For every  2 S N 1 such that   ny < 0 2C˛

lim Uy .y C r /r q1 ! !./

r !0

(3.4.34)

where ! is he solution of problem (3.4.36). For every c 2 .0, 1/, the convergence is uniform in S N 1 \ Œ  ny < c. Recall that ny denotes the unit normal on @ at y pointing outwards. Remark 3.4. Actually, the uniqueness result is valid for ˛ > 2. However, the proof in the case 2 < ˛ < 0 requires a version of the boundary Harnack principle which is beyond the scope of this book. The asymptotic result of part (iii) is also valid for ˛ > 2; see Lemma 3.4.8. The proof of the theorem is based on three lemmas. In the first lemma we establish the result in the case of a half-space.

98

Chapter 3 The boundary trace and associated boundary value problems

Lemma 3.4.7. Assume that  D RN C D Œx1 > 0, 1 < q < qc and ˛ > 2. N For every y 2 @RC there exists a unique very singular solution at y (denoted by Uy ) and 2C˛

U0 .x/ D r  q1 !./

where jxj D r ,  D x=r ,

8x 2 RN C.

(3.4.35)

where ! is a solution of the problem:  ! C !  .  e1 /˛ ! q D 0 !D0 where

  2C˛ 2C˛ C2N , D q1 q1

N 1 on SC N 1 on @SC ,

(3.4.36)

N 1 SC D ¹x 2 S N 1 : x1 > 0º

and e1 is the unit vector in the direction of the x1-axis. Proof. Let U1,0 be defined as in (3.4.31) and let Taq,˛ , a > 0, be the similarity transformation defined in (3.4.4). Since 1 < q < qc , U1,0 is not the trivial solution. The q,˛ function Ta U1,0 is again a solution of (3.4.2) in RN C which vanishes on @ n ¹0º. The mapping u 7! Taq,˛ u is a (1-1) mapping of the set of solutions of (3.4.2) in RN C q,˛ onto itself. Furthermore, this mapping is order preserving. Therefore Ta U1,0 is the q,˛ maximal solution. Thus Ta U1,0 D U1,0, i.e. U1,0 is self-similar. This implies x 2C˛ U1,0 .x/ D a q1 U1,0 8x 2 RN (3.4.37) C a which is equivalent to 2C˛

U1,0 .x/ D r  q1 !./ where jxj D r ,  D x=r , N 1 U1,0 ./ D: !./ 8 2 SC .

8x 2 RN C

(3.4.38)

A straightforward computation shows that ! is a solution of problem (3.4.36). Next we show that ! is the unique positive solution of problem (3.4.36). If ! 0 is 2C˛ any positive solution of this problem then v D r  q1 ! 0 ./ is a solution of (3.4.1) vanishing on @ n ¹0º so that v  U1,0. Hence ! 0  !. Multiplying the equation in (3.4.36) by ! 0 and integrating by parts yields: Z Z   0 ˛ q 0 r!  r! C .  e1/ ! ! dx D  !! 0 dx. N 1 SC

By interchanging the roles of ! and ! 0 we obtain Z Z   0 ˛ 0q r!  r! C .  e1/ ! ! dx D  N 1 SC

N 1 SC

N 1 SC

!! 0 dx.

99

Section 3.4 A problem with fading absorption

Subtracting the two equations yields: Z .  e1/˛ !! 0 .! q1  ! 0q1 /dx D 0. N 1 SC

Since !  ! 0 > 0 we conclude that ! D ! 0 . By (3.4.38), Z q

B1 .0/\Œx1 >0

U1,0.x/˛C1 dx D 1.

Thus U1,0 is a very singular solution at 0. By Theorem 3.4.5 (i), for every k > 0 there exists a (unique) solution of (3.4.1) with boundary data kı0 . By Theorem 3.1.15 u1,0 D lim uk,0 k!1

is the smallest very singular solution at 0. Evidently u1,0 is self-similar. Consequently it satisfies (3.4.38). By the uniqueness result for positive solutions of (3.4.36) we conclude that u1,0 D U1,0 and consequently there exists one and only one very singular solution at 0.  Lemma 3.4.8. Let  be a uniformly C 2 domain and assume that 1 < q < qc and ˛ > 2. Let 0 2 @ and assume that the set of coordinates is positioned so that the hyperplane x1 D 0 is tangent to @ at 0 with the positive x1-axis pointing into the domain. Let u stand for either u1,0 or U1,0 defined as in the proof of Lemma 3.4.7. Then 2C˛

lim r q1 u.x/ D !./

r !0

8 D

x N 1 2 SC r

(3.4.39)

N 1 . uniformly in compact subsets of SC

Proof. For every a > 0, let uak,0 denote the solution of (3.4.1) in a :D a1  with boundary trace kı0 , extended to RN n ¹0º, by setting it zero outside a [ ¹0º. As usual we denote ua1,0 D lim uak,0 k!1

and observe that ua1,0 D Taq,˛ u1,0.

(3.4.40)

If ¹anº is a sequence of positive numbers converging to zero, we can extract a subsequence (still denoted by ¹an º) such that n uak,0 ! vk ,

n ua1,0 !V

uniformly in compact subsets of RN n ¹0º and vk and V are solutions of (3.4.1) in N RN C that vanish on @RC n ¹0º. To verify this assertion, observe that – due to the local Keller–Osserman condition and Proposition 3.4.2 – the sequences are uniformly

100

Chapter 3 The boundary trace and associated boundary value problems

bounded in every compact subset of RN n ¹0º. Therefore linear elliptic estimates (up to the boundary) imply the assertion. Assertion 1. vk D u0k,0 , namely, the solution of (3.4.1) in RN C with boundary trace kı0. Let s > 0 and a,s :D a \ Bs .0/. Denote by vka,s the solution of the problem v C a jvjq1 v D 0 v D kı0 where

in a,s , on @a,s ,

(3.4.41)

a .x/ D dist .x, @a /.

Then

uak,0  vka,s a,s

in a,s .

(3.4.42)

RN C

\ Bs .0/ as a ! 0. Denote by Pa,s the Poisson kernel Keeping s fixed,  ! for  in a,s . Since 1 < q < qc we have Z 1 1N .r 1C˛CN /q dr < 1. 0

Therefore, if a > 0 is sufficiently small then, by Lemma 2.3.3 and Theorem 2.3.4, for every > 0 there exists ı. / > 0 such that for every Borel set E  RN C, Z m.E/ < H) a˛C1Pa,s .x, 0/q dx < ı. /. E\a,s

Since

vka,s .x/  kPa,s .x, 0/ 8x 2 a,s

a standard argument (as in the proof of Theorem 2.3.7) implies that vka,s ! vk0,s where vk0,s is the solution of (3.4.1) in RN C \ Bs .0/ with boundary data kı0 . By (3.4.42), 0,s letting a ! 0, we obtain, vk  vk . Clearly vk0,s ! u0k,0 as s ! 1. Therefore vk  u0k,0 .

(3.4.43)

On the other hand, it is easily verified that vk  u0k,0 . This proves Assertion 1. N Now V is a solution of (3.4.1) in RN C which vanishes on @RC n ¹0º. Furthermore, V  vk D u0k,0 for every k > 0. Therefore V is a very singular solution of (3.4.1) in RN C . By uniqueness – Lemma 3.4.7 – V D U0 . Since the limit is independent of the sequence we conclude that 2C˛

lim ua1,0 .x/ D U0 .x/ D jxj q1 !.x=jxj/ a#0

uniformly in compact subsets of RN C.

(3.4.44)

101

Section 3.4 A problem with fading absorption

N 1 Let S1 be a compact subset of the open half sphere SC and let rN be a positive number such that ¹x : jxj  rN , x=jxj 2 S1 º  .

Then S1  a1  for 0 < a  rN . As the convergence is uniform on S1, (3.4.44) implies 2C˛

lim a q1 u.ax/ D !.x/

a!0

uniformly 8x 2 S1 ,

which is equivalent to (3.4.39). The proof of (3.4.39) in the case u D U1,0 is similar but simpler. In this case we observe that a D Taq,˛ U1,0 U1,0 a is the maximal solution in a vanishing on @a n ¹0º. Therefore U1,0  ua1,0 a and lim infa!0 U1,0  U0 . Hence, by Lemma 3.4.7, (3.4.44) holds with respect to a . U1,0 

Lemma 3.4.9. Let u1 , u2 be two positive solutions of (3.4.1) where q > 1 and ˛ 2 R. Suppose that there exists a constant C > 1 such that u1 < u2  C u1 .

(3.4.45)

Then there exists a solution u such that 1 u1  u < u1 . 1CC Proof. Let  D

1 C C1

(3.4.46)

and put

V D u1  .u2  u1 / D .1 C /u1  u2 . Note that u1  V < u1 and

 q q V D .1 C /u1  u2  ˛ < V q  ˛ .

To verify the last inequality we use the fact that (by convexity)   q V C u2 q V q C u2 q . u1 D  1C 1C Thus V is a supersolution of (3.4.1) and V dominates the subsolution u1 . Therefore there exists a solution u satisfying (3.4.46).  Proof of Theorem 3.4.6. Assume that the set of coordinates is positioned so that y D 0 and the hyperplane x1 D 0 is tangent to @ at 0 with the positive x1 -axis pointing into the domain.

102

Chapter 3 The boundary trace and associated boundary value problems

As usual we denote by U1,0 the maximal solution of (3.4.1) vanishing on @ n ¹0º and put u1,0 :D lim ukı0 . k!1

Both are very singular solutions and by Theorem 3.1.15, u1,0 is the minimal such solution. By Lemma 3.4.8, for every ˇ > 0 there exists a constant cˇ such that 2C˛

2C˛

cˇ1 jxj q1 1 .x/  u1,0 .x/  U1,0.x/  cˇ jxj q1 1 .x/

(3.4.47)

in the truncated cone Eˇ :D ¹x 2  : jxj < ˇ.x/, .x/ < r0º where r0 is the first component of the C 2 characteristic of . The constant does not depend on y but only on the factors mentioned in Theorem 3.4.6 (ii). Next we prove that inequality (3.4.47) also holds in .Br0=4 .0/ \ / n Eˇ . Let  2 @ n ¹0º and denote r ./ :D Put

q1

h1 D  ˛ u1,0 ,

1 min.jj, r0 /. 8

q1

h2 D  ˛ U1,0 ,

Lm D  C hm , m D 1, 2

so that L1 u1,0 D 0,

L2U1,0 D 0.

(3.4.48)

Denote by M the maximum of the norms of r ./ hm in C . \ B4r . / .// (m D 1, 2). Since, by the assumption ˛  0, Proposition 3.4.4 implies that M is bounded by a constant M independent of . Therefore, in view of our assumption on , Corollary 1.6.3 and (3.4.48) imply that there exists a constant C independent of  such that 2

1 .x/ w.x/ .x/  C C r ./ w.  r ./n / r ./

1

8x 2  \ T ., r .//

(3.4.49)

for w D u1,0 and w D U1,0.1 Note that z D   r ./n H) .z/ D jz  j D r ./, r ./ < jzj < 10r ./. Therefore .z/ < jzj < 10.z/ and, if jj < r0 so that .z/ D r ./ D jj=8, we have 7 9 jj  jzj  jj. 8 8 1

The assumption ˛  0 is used at this point in order to guarantee that M is uniformly bounded. This, in turn, is used in order to apply Corollary 1.6.3.

103

Section 3.4 A problem with fading absorption

Thus z 2 Eˇ (with ˇ D 10) and therefore (3.4.47) holds at the point z. By (3.4.49), 1 .x/ .x/ u0,1 .z/  u0,1 .x/  U0,1 .x/  C U0,1 .z/ C r ./ r ./ for every x 2  \ T ., r .//. Hence applying (3.4.47) to u0,1 .z/ and U0,1 .z/ we obtain 1 .x/  2C˛ 1 .x/  2C˛ 1 jzj q1 .z/  u1,0 .x/  U1,0.x/  C1 jzj q1 .z/ C1 r ./ r ./ for every x 2  \ T ., r .//, C1 D C cˇ , ˇ D 10. This implies that, if jj < r0, 2C˛ 2C˛ 1 .x/jj q1 1  u1,0.x/  U1,0 .x/  C2.x/jj q1 1 , C2

for every x 2  \ T ., r .// and r ./ D jj=8. Finally, for jj < r0 and x 2  \ T ., r .// we have 1 3 jj < jxj < jj. 2 2 Therefore, if x 2  \ Br0 =4 .0/ and .x/ < jxj=16, 2C˛ 1  2C˛ 1 jxj q1 .x/  u1,0.x/  U1,0 .x/  cjxj q1 1 .x/ c

(3.4.50)

where c D 2C2. Combining this estimate with (3.4.47) we conclude that (3.4.50) holds for every x 2  \ Br0=4 ./. We turn now to the proof of uniqueness of the very singular solution, first assuming that  is bounded. By Hopf’s lemma, there exists c  D c  .r1/ > 0 such that 1 .x/  u1,0 .x/  U1,0.x/  c  .x/ c for every x 2  n Br0=4 .0/. Therefore, in this case, (3.4.50) – with a constant c depending on the parameters mentioned in assertion (ii) – holds for every x 2 . Hence, there exists a positive constant C such that, u1,0  U1,0  C u1,0 in . Consequently, by Lemma 3.4.9, either u1,0 D U1,0 or there exists a solution v of (3.4.1) such that 1 u1,0  v < u1,0. 1CC

(3.4.51)

104

Chapter 3 The boundary trace and associated boundary value problems

It follows that v is a very singular solution at 0. But, by Theorem 3.1.15, u1,0 is the smallest very singular solution at 0. Therefore (3.4.51) holds. Finally we prove uniqueness in the case when  is unbounded. Let ¹n º be an increasing sequence of bounded C 2 domains as in the proof of Corollary 3.3.3. Let U0n be the unique very singular solution of (3.4.1) in n with singularity at 0. This sequence increases and U :D lim U0n is a very singular solution at 0 in . If W is another such solution then U0n  W in n . Consequently, U is the smallest very singular solution at 0. Put Fn D @n \ , Fn0 D @n n Fn . Let wn,k be the solution of (3.4.1) in n such that ´ 0 if x 2 Fn0 wn,k .x/ D k if x 2 Fn . Put wn D lim wn,k ,

w D lim wn . n!1

k!1

(Note that ¹wn º decreases.) Then w is a solution of (3.4.1) in  which vanishes on @. Let E be a compact subset of  and let > 0. By Proposition 3.4.2 there exists ˇ 2 .0, r0=2/ such that w.x/ 

8x 2  : .x/ < ˇ.

We may and shall assume that ˇ is sufficiently small so that E  Dˇ . Since .x/ > ˇ in Dˇ , g.x, t /  ˇ ˛ t q

8x 2 Dˇ , t > 0.

Let zn be the maximal solution of the equation z C ˇ ˛ jzjq1 z D 0 in Dˇ \ n which vanishes on @.Dˇ \ n / n Fn . Then zn .x/  c.ˇ/dist .x, Fn / q1 . 2

The constant c.ˇ/ depends on ˇ, q, ˛, but not on n. Note that zn is a supersolution of the equation (3.4.1) in Dˇ \ n . Therefore C zn is a supersolution of this equation and wn,k  C zn on @.Dˇ \ n / 8k 2 N.

105

Section 3.4 A problem with fading absorption

Therefore wn .x/  C c.ˇ/dist .x, Fn \ Dˇ /2=.q1/

8x 2 Dˇ \ n .

Hence w  in E. Since is arbitrary we conclude that w D 0. The function .U1,0  wn /C is a subsolution of (3.4.1) in n which vanishes on @n n ¹0º. Therefore, .U1,0  wn /C  U0n and, as wn ! 0, U1,0  U . Since U is the smallest while U1,0 is the largest very singular solution at 0, uniqueness follows. Statement (iii) follows from Lemma 3.4.8.  We conclude with the following existence and uniqueness result, which provides a complete characterization of the positive solutions of (3.4.1) in terms of their boundary traces, in the subcritical case. Theorem 3.4.10. Let  be a (possibly unbounded) domain, uniformly of class C 2 and assume 1 < q < qc D .N C ˛ C 1/=.N  1/, 0  ˛. Then problem (3.4.3) possesses a unique solution for every N 2 Breg. Furthermore, if F is a closed subset of @ then UF – the unique solution with boundary trace 1F – satisfies the inequality 2C˛ 2C˛ 1 dist .x, F / q1 1 .x/  UF .x/  cdist .x, F / q1 1 .x/ c

(3.4.52)

for every x 2  such that dist .x, F / < r0=4. The constant c depends only on the parameters mentioned in Theorem 3.4.6 (ii). Remark 3.5. Recall that, in the case of an unbounded domain, the singular set of a solution is closed but not necessarily compact. Proof. Let ., F / be a representation of N as in Definition 3.1.13. Existence follows from Corollary 3.3.3 because, in the subcritical case, conditions (i) and (ii) are satisfied by any measure in Breg . Next we show that, for every compact set F  @, the maximal solution UF is the unique solution of problem (3.4.3) with trace N D 1F . We begin by constructing the minimal solution with this trace. Let ¹x nº  F be a sequence dense in F and put k D k

k X j D1

ıx j .

106

Chapter 3 The boundary trace and associated boundary value problems

Let vk be the unique solution of (3.4.1) with boundary trace k . The sequence ¹k º is increasing and we denote VF D lim vk . Then VF  Uxn

n D 1, 2, : : : .

where Uy is the unique very singular solution at y. Therefore ¹x n º  S.VF /  F . As S.VF / is a closed set it follows that S.VF / D F . By Theorem 3.1.15, if v is a positive solution of (3.4.1) such that S.v/ D F then v  Uy Therefore v  vk and finally

8y 2 F .

v  VF .

Thus VF is the minimal solution with boundary trace 1F . It remains to prove that, if UF is the maximal solution vanishing on @ n F , then UF D VF .

(3.4.53)

If x 2  and dist .x, F / < r0 =4, pick y 2 F such that jx  yj D dist .x, F /. Then, by (3.4.33), 2C˛ 1 jx  yj q1 1 .x/  Uy .x/  VF .x/. c

(3.4.54)

Let  2 @ n F be such that dist ., F / < r0=4 and pick y 2 F as before. By the same argument used in proving (3.4.50) we obtain 2C˛

UF .x/  cjx  yj q1 1 .x/ for every x 2  \ Br0=4 .y/ such that .x/ < jx  yj=16. On the other hand, if .x/  jx  yj=16 then, by the Lemma 3.4.1, 2C˛

2C˛

UF .x/  C.N , q/jx  yj q1  16C.N , q/jx  yj q1 1 .x/. Hence, 2C˛

UF .x/  c 0 dist .x, F / q1 1 .x/,

(3.4.55)

for every x 2  such that dist .x, F / < r0 =4. Combining (3.4.54) and (3.4.55) we obtain, 2C˛ 2C˛ 1 dist .x, F / q1 1 .x/  VF .x/  UF .x/  cdist .x, F / q1 1 .x/. c

107

Section 3.5 Notes

It follows that there exists C > 0 such that UF  C VF in the set ¹x 2  : dist .x, F / < r0 =4º. If  is bounded then, by Hopf’s lemma, it follows that UF =VF is bounded in . As VF is the minimal solution with boundary trace 1F , Lemma 3.4.9 implies that UF D VF . If  is unbounded, uniqueness is proved as in Corollary 3.3.3. We recall that, if wn is constructed as in that proof, the fact that wn ! 0 is established by the argument given in the proof of uniqueness for Theorem 3.4.6. 

3.5 Notes A definition of a boundary trace for arbitrary positive solutions, equivalent to the one presented in Section 3.1, was introduced by M. Marcus and L. Véron in [75] and [77] in the case of power nonlinearities, assuming that the domain is a ball in RN , N  2. In the two-dimensional case, a related definition applying to positive solutions of u C u2 D 0 in a disk, expressed in probabilistic terms, was previously introduced by J.F. LeGall [65] and [66]. The relation between the analytic definition of [75] and the probabilistic definition of [65] has not been directly investigated. But a posteriori, based on existence and uniqueness results for the corresponding boundary value problems, it can be deduced that the two are equivalent. The definition introduced in [75] was extended to a general class of semilinear elliptic equations with absorption, in domains of class C 2, in [80]. E.B. Dynkin [39] referred to this notion of trace as the ‘rough trace’. Most of Subsection 3.1 extends results of [80] in which the authors treated the case g.x, t / D .x/˛ jt jq1 t ,

1 < q < .N C ˛ C 1/=.N  1/.

Theorem 3.3.1 is taken from [73] and [86]. (In the latter the result was adapted to Lipschitz domains.) The results contained in Subsection 3.4 are based on [80]. A general class of semilinear equations admitting a similarity transformation is treated in [17]. Proposition 3.4.2 is based on a construction of Y. Du and Z. Guo [38].

Chapter 4

Isolated singularities

This chapter is devoted to the study of singular solutions of semilinear elliptic equations in a domain of RN . The properties of solutions under consideration are essentially the a priori estimates and the precise description of isolated singularities.

4.1 Universal upper bounds The first striking result concerns universal bounds of solutions of semilinear elliptic equations with absorption. If g.x, r / is a superlinear function in the variable r , any function u satisfying a differential inequality of the type u C g ı u  0

(4.1.1)

in a domain  of RN admits, on any compact subset K of , an upper bound which depends only on N , the distance from K to c and some constants associated to the growth of g. The existence of a local universal upper bound has been established for a large variety of second order quasilinear equations, of elliptic and parabolic type. We recall that .x/ is the distance from x 2  to @.

4.1.1 The Keller–Osserman estimates In Chapters 2 and 3 we have already introduced the class of functions G0 ./ (see .2.1.2/) and a Keller–Osserman condition (see Definition 3.1.9) adapted to deal with trace problems and power nonlinearities. We give below a more general form adapted to particular exponential type nonlinearities. Furthermore, no monotonicity is needed to obtain local universal bounds. Definition 4.1.1. We denote by GQe D GQe ./ the class of functions g defined in  R which satisfy the following conditions: .a/

g.x, / 2 C.R/, g.x, ./ > 0

.b/

g., t / 2

L1loc ./

8t 2 R.

a.e. in 

(4.1.2)

A function g 2 GQe satisfies the extended Keller–Osserman condition if there exists a positive non-decreasing function h satisfying g.x, r /  h.r / for all .x, r / 2 . n F / R where F   has zero Lebesgue measure and .3.1.14/ holds for every a 2 R.

109

Section 4.1 Universal upper bounds

Theorem 4.1.2. Let  be a domain in RN and g 2 GQe satisfying the extended Keller– Osserman condition. There exists a non-increasing function fQg defined on RC with the limits lim fQg ./ D1,

.i/

!0

(4.1.3)

lim fQg ./ D  1,

.ii/

!1

such that any u 2 L1loc ./ with g.., u/ 2 L1loc ./ which satisfies (4.1.1) in the sense of distributions and verifies the following upper estimate u.x/  fQg ..x//

for almost all x 2 .

(4.1.4)

Proof. By Lemma 1.5.3, u  g.., u/ is a positive distribution. There exists a nonpositive bounded Borel measure  such that u C g.., u/ D 

in . 1,p

Since u, as well as g ı u, is locally integrable in , it belongs to Wloc ./ for any p 2 Œ1, N=.N  1//. Let x0 2  and R0 D .x0/. For 0 < R < R0 , we set D D BR .x0 / and @D D @BR .x0/. Then u admits a Sobolev trace ub@D on @D which, in particular, belongs to L1.@D/. Thus u coincides with the solution U of U C g ı u D  U D ub@D

in D :D BR .x0/ on @D.

(4.1.5)

For k > 0, we denote by v D vk,R the solution of v D h ı v vDk

in D, on @D.

(4.1.6)

Existence and uniqueness are standard because h is monotone, and vk,R D v depends only on r D jx  x0 j. Therefore v 0 .0/ D 0 and v is uniquely determined by v0 :D v.0/ since it satisfies v 00  Nr1 v 0 C h ı v D 0 in .0, R/ (4.1.7) v 0 .0/ D 0, v.0/ D v0 . This yields

Z

Z

r

v.r / D v0 C

t

t

1N

0

h.v.s//s N 1 dsdt .

(4.1.8)

0

Hence v is increasing and (4.1.8) is linked to (4.1.6) by Z k D v.r / C

Z

R

t r

t

1N 0

h.v.s//s N 1 dsdt .

(4.1.9)

110

Chapter 4 Isolated singularities

By the maximum principle, .k, R/ 7! vk,R is increasing in k and decreasing in R. We denote (4.1.10) v1 .r / D lim lim vk,R .r /, R!R0 k!1

a quantity which could be finite or infinite. We set w D u  v. Since   0 and g ı u  h ı u, there holds wC C .h ı u  h ı v/sgnC.u  v/  0

in D

by Kato’s inequality. Since h is non-decreasing, wC is a subharmonic function in D. We apply .1.5.20/ in Proposition 1.5.9, with D 0D , to the solution of  D 1

D0 and we get

in D on @D,

Z

Z D

wC dx  c

@D

.u  k/C dS

8k > 0.

(4.1.11)

By the dominated convergence theorem, the right-hand side of (4.1.10) tends to 0 when k ! 1. This implies u.x/  v1 .jx  x0 j/

for almost all x 2 .

(4.1.12)

It remains to show that v1 .r / is finite for 0  r < R0 . By (4.1.9) Z r Z r r 0 1N N 1 1N h.v.s//s ds  r h.v.r // s N 1 ds D h.v.r //, v .r / D r N 0 0 thus

  N 1 0 N 1 1 v .r / D  v .r / C h.v.r //  1  h.v.r // D h.v.r //. r N N 00

Using the differential equation satisfied by v and the positivity of v 0 , we get 1 h ı v  v 00  h ı v. N

(4.1.13)

Multiplying by v 0 and integrating over .0, r /, we obtain 2 .H.v.r //  H.v0 //  .v 0 .r //2  2.H.v.r //  H.v0 //, N R where H./ D 0 h.s/ds. Hence r p  N

Z

v.r / v0

ds p  r. 2.H.s/  H.v0 //

(4.1.14)

(4.1.15)

111

Section 4.1 Universal upper bounds

If we use the fact that h is positive and non-decreasing and estimate .3.1.14/ holds, it implies Z 1 ds <1 A.v0 / D p 2.H.s/  H.v0 // v0 for any v0 2 R. Hence (4.1.15) can be written in the form Z v.r / p ds  r  N A.v0 /. p (4.1.16) 2.H.s/  H.v0 // v0 p Therefore, there exists r D R.v0 / 2 .0, N A.v0 / such that the maximal solution of (4.1.7), still denoted by v, satisfies v.R.v0 // D 1. By uniqueness and continuous dependence of solutions of differential equations, v0 7! R.v0 / is decreasing and continuous. Furthermore p (4.1.17) A.v0 /  R.v0 /  N A.v0 / by (4.1.15) and lim A.v0 / D 1 and lim A.v0 / D 0.

v0 !1

v0 !1

(4.1.18)

By (4.1.17), v0 7! R.v0 / inherits the same limits. If fQg is the inverse function of R, it is a decreasing homeomorphism from .0, 1/ onto .1, 1/ which verifies (4.1.3) (i)–(ii), and by (4.1.12), u.x/  f ..x// for almost all x 2 .  In a variant of the previous theorem a positive upper estimate for a weak solution of (4.1.1) is proven under a slightly different Keller–Osserman condition (see Definition 3.1.9). We introduce a variant of the class G0./ Definition 4.1.3. A function g defined in  R belongs to the class GQ0 D GQ0 ./ if it satisfies .a/ g.x, / 2 C.R/, g.x, t /  0 if t  0, (4.1.19) .b/ g., t / 2 L1loc ./ 8t 2 R. It satisfies the positive Keller–Osserman condition if there exists a non-negative nondecreasing function h satisfying g.x, r /  h.r / for all .x, r / 2 . n F / RC where F   has zero Lebesgue measure, and .3.1.14/ holds for every a > 0. Theorem 4.1.4. Let   RN and g 2 GQ0 . If we assume that g satisfies the positive Keller–Osserman condition, there exists a non-increasing non-negative function fQ defined on RC with the limits .i/ lim fQ ./ D 1, !0

.ii/

lim fQ ./ D 0,

!1

(4.1.20)

112

Chapter 4 Isolated singularities

such that, if u 2 L1loc ./ with g ı u 2 L1loc ./ is a subsolution of (4.1.1), u.x/  fQ ..x//

for almost all x 2 .

(4.1.21)

Proof. We use Kato’s inequality and Lemma 1.5.3 as in the proof of Theorem 4.1.2. The function v D vk,R is defined by the same equation, however, it is non-negative in D for any k > 0, since we can always assume h.0/ D 0. The function v1 , which is larger than u almost everywhere in D, is defined by formula (4.1.10). The relations (4.1.16)–(4.1.17) remain unchanged with the restriction that v0 be non-negative. Formula (4.1.18) is changed into lim A.v0 / D 1 and lim A.v0 / D 0, v0 !1

v0 !0

(4.1.22)

and v0 7! A.v0 / is a continuous decreasing function from .0, 1/ into .0, 1/. Thus its inverse function fQ satisfies (4.1.20) (i)–(ii). Clearly (4.1.21) follows.  If (4.1.1) is replaced by the equation u C g ı u D 0,

(4.1.23)

a two-side estimate of solutions can be derived under the global Keller–Osserman condition. We recall that we define g  from g by g  .x, t / D g.x, t / for all .x, t / 2  R. Theorem 4.1.5. Let   RN . Assume g and g  belong to GQ0 and satisfy the positive Keller–Osserman condition. Then there exist two non-increasing non-negative functions fQ and fQ defined on RC with the limits (i)

lim fQ ./ D lim fQ ./ D 1,

!0

!0

lim fQ ./ D lim fQ ./ D 0, (4.1.24)

!1

!1

such that any u 2 L1loc ./ with g ı u 2 L1loc ./, solution of (4.1.23) in  satisfies f ..x//  u.x/  f  ..x//,

8x 2 .

(4.1.25)

Proof. From the assumption there is a non-negative non-decreasing function h0 satisfying g  .x, r /  h0 .r / for all .x, r / 2 . n F 0 / RC where F 0   has zero Lebesgue measure, and .3.1.14/ holds for every a0 > 0. We apply Theorem 4.1.4 to the function uQ D u, since uQ D g  ı uQ in ,  and fQ is obtained in the same way as fQ . Remark 4.1. Similarly as in Definition 3.1.9, it is easy to give local versions of the extended, positive and global Keller–Osserman conditions. In many applications g.x, t / D `.x/g.t / and local estimates of the minorant functions h and h0 are given explicitly.

113

Section 4.1 Universal upper bounds

4.1.2

Applications to model cases

Example 1. If g.r / D e ˛r for some ˛ > 0, Theorem 4.1.2 applies. Any solution of u C e ˛u  0

(4.1.26)

  1 3 2  ln ˛ C N . u.x/  ln ˛ .x/ 2

(4.1.27)

in domain   RN satisfies

As a consequence there exists no solution of (4.1.26) if  D RN . Consider now u C a.x/e ˛u  0

(4.1.28)

where a.x/ > 0. If G is a subdomain of  and minG a.x/ D aG > 0, v D u  ˛ 1 ln aG satisfies (4.1.26). It follows from (4.1.27) that ! 1 3 1  ln ˛ C N . (4.1.29) u.x/  ln 2 ˛ aG @G .x/ 2 In particular, if we take G D Br .x/ with 0 < r < .x/, any solution of (4.1.28) in  satisfies ! 1 1 3 u.x/  ln (4.1.30)  ln ˛ C N ,  2 ˛ a .x/@ .x/ 2 where

a .x/ D max. 2

min

B.x/ .x/

a.y/ : 0 <  < 1/.

(4.1.31)

Example 2. If q > 1, any solution of the inequality

in  verifies

u C jujq1 u  0

(4.1.32)

u.x/  CN ,q ..x//2=.q1/ .

(4.1.33)

A consequence is a Liouville type result for Emden-Fowler equations: if u is a solution of u C jujq1 u D 0, (4.1.34) in RN , it is identically 0. Similarly, if u is a solution of u C a.x/jujq1 u  0

(4.1.35)

in , where a.x/ > 0, it satisfies  1=.q1/ , u.x/  CN ,q a .x/ 2.x/ where a is given by (4.1.31).

(4.1.36)

114

Chapter 4 Isolated singularities

Example 3. Let u satisfy u C jujq1 u  jujp1 u  0 (4.1.37)   p qp on RC, g.r C1/  cr q for some 0  p < q. Since g.r / :D r q r p  .pq/ pq for some c > 0, u verifies u.x/  CN0 ,q ..x//2=.q1/ C 1.

(4.1.38)

4.2 Isolated singularities Let q > 1, ˛ 2 R and N  2. If ˛  2, it is a consequence of (4.1.36) that any solution of (4.2.1) u C jxj˛ jujq1 u D 0 in BR n ¹0º is locally bounded, thus we only consider the case ˛ > 2. Separable solutions of (4.2.1) in RN n ¹0º under the form u.x/ D r ˇ where r D jxj, may exist 1=.q1/ with ˇ D .2 C ˛/=.q  1/ and  D N ,q,˛ with    2q C ˛ N C˛ 2C˛ N , q > . (4.2.2) N ,q,˛ D q1 q1 N 2 Thus N ,q,˛ > 0 only if 1 < q < .N C ˛/=.N  2/ when N  3, or if ˛ > 2, when N D 2.

4.2.1 Removable singularities We give below a basic result on removability of isolated singularities. Theorem 4.2.1. Assume ˛ > 2 and q  qe,˛ :D .N C ˛/=.N  2/. Then any solution u of (4.2.1) in  n ¹0º can be extended to a locally bounded L1 weak solution uQ in . If ˛  0, uQ 2 C 2./. Proof. Let R > 0 such that B2R  . Then there holds from (4.1.31) and (4.1.36) ju.x/j  CN ,q,˛ jxj.2C˛/=.q1/

8 0 < jxj  R.

(4.2.3)

By Kato’s inequality, uC satisfies q

uC C jxj˛ uC  0

(4.2.4)

in the L1-weak sense in B2R n ¹0º. Let n 2 C 1 .RN / such that, n  0, n .x/ D 0 if jxj  n1 , n.x/ D 1 if jxj  2n1 , jrn j  C n, jn j  C n2 . If  2 Cc1 .BR / verifies 0    1 and has value 1 in BR=2 and if n D n , it holds that Z q .uC n C jxj˛ uC n /dx  0. (4.2.5) BR

115

Section 4.2 Isolated singularities

But n D n C n C 2r.rn, thus Z Z Z 2 juC nj dx  jj nuC dx C C.n C 2nkrkL1 / BR

BR

By (4.2.3), .n C 2nkrkL1 /

Z

2

since

2C˛ q1

n1 jxj2n1

uC dx.

uC dx  C n.2C˛/=.q1/C2N  C 0 as n ! 1,

C 2  N  0. Furthermore, with q 0 D q=.q  1/, Z

Z BR

n1 jxj2n1

jj n uC dx 

q0

BR

˛q 0 =q

jj jxj

1=q 0 Z n dx

BR

q jxj˛ uC n

1=q .

0

Since q  qe,˛ , jxj˛q =q 2 L1.BR /, therefore Z BR

q jxj˛ uC n dx

Z C

BR

q jxj˛ uC n dx

1=q Z

 .n2 C 2nkrkL1 /

n1 jxj2n1

uC dx. (4.2.6)

q

By Fatou’s lemma, jxj˛ uC 2 L1.BR /. It also shows that u 2 L1.BR /. Next we assume that  2 Cc1 .BR / is non-negative. From (4.2.5) we derive Z Z ˛ q 2 .uC  C jxj uC /ndx  C.n C 2nkrkL1 / uC dx. n1 jxj2n1

BR

By Hölder’s inequality, Z Z uC dx  n1 jxj2n1

1=q Z

jxj˛ uqC dx

n1 jxj2n1

˛q 0 =q

jxj

n1 jxj2n1

(4.2.7) 1=q 0 dx

 C nN C.˛CN /=q o.1/. Since q  qe,˛ , N C .2 C ˛/=.q  1/  2. Therefore the right-hand side of (4.2.7) goes to zero as n ! 1. By the dominated convergence theorem uC is a subsolution in BR . Therefore it is bounded by Theorem 4.1.4. Similarly u is bounded in BR . Finally, if ˛  0, u is C 2 by standard regularity theory.  Remark 4.2. Two alternative proofs of this results exist. They are based either upon similarity transformations or the energy method. Both rely on the fact that there exists no nontrivial separable solution of (4.2.1) in RN n¹0º . We shall develop these methods in the next sections in slightly different contexts. As a consequence, we derive the following result:

116

Chapter 4 Isolated singularities

Corollary 4.2.2. Let , a domain of RN (N  3) containing 0,  :D  n ¹0º and g 2 GQ0./, satisfy lim inf r qe,˛ jxj˛ g.x, r / > 0 and lim inf r qe,˛ jxj˛ g.x, r / < 0, r !1

r !1

(4.2.8)

for some ˛ > 2, essentially locally uniformly with respect to x. If u is an L1 weak solution of u C g ı u D 0 (4.2.9) in  , it remains locally bounded in . If we assume moreover that for any 2 ./, g.., ..// 2 L1loc ./, then u is an L1 weak solution in . L1 loc Proof. Since .4.2.8/ holds essentially uniformly with respect to x, there exist a > 0 and m  0 such that g.x, r /  ajxj˛ r qe,˛ for r  m and x 2 BR n F where R > 0 is such that B2R   and F has zero Lebesgue measure. Since .u  m/ C g ı u D 0, there holds by Kato’s inequality .u  m/C C ajxj˛ sgnC .u  m/Cuqe,˛  0 in the sense of distributions in BR n ¹0º. Thus q

.u  m/C C ajxj˛ .u  m/Ce,˛  0, and by the maximum principle .u  m/C is bounded from above. In the same way, u is bounded from below, thus u 2 L1 .BR /. If the second assumption is verified, g.., u..// 2 L1loc ./. Therefore, the argument already used in the proof of Theorem 4.2.1 implies that u is an L1 weak solution. 

4.2.2 Isolated positive singularities If 1 < q < qe,˛ there exists an explicit singular radial solutions to (4.2.1) in RN n ¹0º, but this is not the only positive singular solution in this domain. Let N be the Newtonian kernel in RN n ¹0º defined by 8 1 ˆ ˆ jxj2N if N ¤ 2 < N.N  2/!N (4.2.10) N .x/ D ˆ ˆ : 1 ln.jxj1 / if N D 2. 2 where !N is the volume of the unit ball. Theorem 4.2.3. Assume that  is a bounded domain in RN , N  2, containing 0 with a smooth boundary, ˛ > 2 and 1 < q < qe,˛ . Then for any k > 0 there exists a unique function uk 2 C 2 . n ¹0º/ \ C. n ¹0º/, solution of (4.2.1) in  n ¹0º, vanishing on @ and satisfying uk .x/ D k. x!0 N .x/ lim

(4.2.11)

117

Section 4.2 Isolated singularities

Proof. By scaling we can assume   BR  B1 . For > 0, we denote by u :D uk, the solution of u C jxj˛ jujq1 u D 0 uD0 u D kN . /

in  n B  on @ on @B .

(4.2.12)

Let GŒf  D G Œf  be the Green potential of a given function f . By the maximum principle 0  uk, .x/  kG .x, 0/ in  n B . Then uk, .x/  kG .x, 0/  k q G Œjxj˛ .G .x, 0//q .

(4.2.13)

Moreover . , k/ 7! uk, is increasing and Z Z q jxj˛ uk, dx  k q jxj˛ .G .x, 0//q dx < 1, 0 nB



this last integral being finite since 1 < q < qe,˛ . Since G .., 0/ vanishes on @, there exists c :D c./ > 0 such that 0 < N .x/  G .x, 0/  c. q Because GBR Œjxj˛ N  is radial with respect to 0, G Œjxj˛ .G .., 0//q  by a bounded function, there

integrable in BR and differs from holds

G Œjxj˛ .G .., 0//q .y/ D 0. y!0 N .y/ lim

(4.2.14)

By (4.2.13), uk :D lim!0 uk, is bounded from below and satisfies (4.2.13) in n¹0º. Then estimate (4.2.11) follows. For proving uniqueness, we consider two solutions u and u0 satisfying (4.2.11). Both functions are positive by the maximum principle and w D u  u0 satisfies w C d.x/w D 0 where d  0 is expressed by 8 q 0q < jxj˛ u .x/  u .x/ if u.x/ ¤ u0 .x/ u.x/  u0 .x/ d.x/ D : 0 if u.x/ D u0 .x/. Because

w.x/ D 0, x!0 N .x/ for any > 0 the function D w  N is negative near 0 and @. Therefore Z   2 jr C j2 C d C dx D 0, lim



and C D 0. Thus w  0 if ! 0. Uniqueness follows by permuting u  and u0 .

118

Chapter 4 Isolated singularities

If  is replaced by RN the following variant of Theorem 4.2.3 holds. Theorem 4.2.4. Assume N  2, ˛ > 2 and 1 < q < qe,˛ . Then for any k > 0 there exists a unique function uk 2 C 2 .RN n ¹0º/, solution of (4.2.1) in RN n ¹0º satisfying (4.2.11). the solution constructed in Theorem 4.2.3, then the Proof. If we denote by u k is increasing by the maximum principle. Because u .x/  map  7! u k k .2C˛/=.q1/ by (4.1.33), we can take  D BR and let R tend to infinity. Then CN ,q jxj BR uk :D limR!1 uk satisfies the equation and the above decay at infinity. Moreover R .x/  kN .x/ C CN ,q uB k

in B1 , for some c with jcj D 1. The monotonicity with respect to R implies that uk satisfies (4.2.11). For uniqueness, we consider two solutions u and u0 ; for > 0, .1 C /u C is a supersolution which dominates u0 at infinity and at 0, thus, by the maximum principle u0  .1 C /u C . Letting ! 0, u0  u. Similarly u  u0 .  Another useful formulation of the above result is the following: Theorem 4.2.5. Assume that , ˛ and q are as in Theorem 4.2.3 and let k > 0. Then the following two assertions are equivalent: (i) u 2 C 2 . n ¹0º/ \ C. n ¹0º/ is a solution of (4.2.1) in  n ¹0º which vanishes on @ and satisfies (4.2.11), (ii) u 2 Lq .; jxj˛ dx/ and for any  2 C02./, there holds Z   u C jxj˛ jujq1 u dx D k.0/.

(4.2.15)



Proof. The implication (i) H) (ii) is based upon the next statement: If u satisfies (i), there holds k x . (4.2.16) lim jxjN 1 ru.x/ D  x!0 N !N jxj If N > 2, we set u` .x/ D `N 2 u.`x/ for ` > 0. Then u` C `N C˛q.N 2/ jxj˛ .u` /q D 0 in ` n ¹0º where ` D `1 . Since 0  u.x/  kN .x/, u` satisfies the same estimate in ` . By the Arzela–Ascoli theorem and elliptic equations regularity theory, 1 the set ¹u` º is relatively compact in the Cloc -topology of RN n¹0º. Furthermore, since N C ˛  q.N  2/ > 0 any limit function v of a subsequence ¹u`n º, when `n ! 0, is a non-negative harmonic function in RN n ¹0º. Since u` .x/ ! kN .x/ uniformly on

119

Section 4.2 Isolated singularities

1 B2 n B1=2, it follows that u` ! kN in the Cloc -topology of RN n ¹0º. This implies ` (4.2.16). If N D 2 we denote u .x/ D u.`x/  k2 .`/ with ` 2 .0, 1/. Then

u` C `2C˛ .u` C k2 .`//q D 0 in ` n ¹0º. If > 0 there exists ı > 0 such that .k  /2 .x/  u.x/  k2 .x/ for 0 < jxj  ı. Therefore  2 .`/ C .k  /2 .x/  u` .x/  k2 .x/

8x 2 Bı=` n ¹0º.

From the equation satisfied by u` and the above estimate, the set ¹u` º is relatively 1 compact in the Cloc -topology of R2 n ¹0º. Thus u` ! k2 in this topology, as ` ! 0. Therefore k x `ru.`x/ !  . 2 jxj2 This implies (4.2.16) by taking jxj D 1. Next we prove that (ii) holds. For > 0 and  2 C02 ./ with b@ D 0, there holds by Green’s formula Z Z Z   u C jxj˛ uq  dx D  un dS C un dS. (4.2.17) nB

@B

@B

By .4.2.16/, the right-hand side of .4.2.17/ converges to k.0/. Thus (4.2.15) holds by the dominated convergence theorem. For (ii) H) (i) it is enough to prove uniqueness. If u0 2 Lq ./ is another solution and w D u  u0 and  2 C02 ./, there holds Z   w C jxj˛ .jujq1 u  ju0 jq1 u0 / dx D 0. 

Since f :D jujq1 u  ju0 jq1 u0 2 L1 .; jxj˛ dx/, it follows from Kato’s inequality that, for any  2 C02 ./, there holds Z .jwj C f sgn.w// dx  0. 

If we take in particular  D , the solution of  D 1 D0 we obtain w D 0.

in  on @, 

Remark 4.3. Using the same construction as in Theorem 4.2.4 it is easy to prove that q Theorem 4.2.5 is valid if  is replaced by RN . In such a case, u 2 Lloc .RN / and C02 ./ is replaced by Cc2 .RN /.

120

Chapter 4 Isolated singularities

Theorem 4.2.6. Assume that ˛ > 2, 1 < q < qe,˛ and  is either a smooth bounded domain containing 0 or RN . For k > 0, let uk be the solution constructed in Theorem 4.2.3 or Theorem 4.2.4. Then k 7! uk is increasing, and u1 D limk!1 uk is a solution of (4.2.1) in  n ¹0º which vanishes on @ and satisfies lim jxj.2C˛/=.q1/ u1 .x/ D N ,q,˛ . 1=.q1/

x!0

(4.2.18)

Furthermore, u1 is the unique function in C 2 .n¹0º/\C. n¹0º/ solution of (4.2.1) in  n ¹0º, which satisfies (4.2.18) and vanishes on @ if  ¤ RN . Proof. We recall that N ,q,˛ , defined in (4.2.2), exists since 1 < q < qe,˛ . Furthermore, by (4.2.11) and the maximum principle, .2C˛/=.q1/ uk .x/ < 1=.q1/ :D Us .x/, N ,q,˛ jxj

(4.2.19)

for any k > 0, and Us is a positive solution of (4.2.1) in RN n ¹0º. Uniqueness and (4.2.11) implies that k 7! uk increases. Therefore lim uk :D u1  Us ,

k!1

(4.2.20)

and u1 2 C 2 . n ¹0º/ \ C. n ¹0º/ vanishes on @ and satisfies (4.2.1) in  n ¹0º. For r > 0, we set (4.2.21) Trq,˛ Œ .x/ D r .2C˛/=.q1/ .r x/. q,˛

Clearly 7! Tr Œ  is continuous with respect to the pointwise convergence of q,˛ functions. Furthermore Tr preserves equation (4.2.1) but the domain  is replaced 1 by r :D r . If  D RN , uniqueness in Theorem 4.2.4 implies that uk is radial and Trq,˛ Œuk  D ukr .2C˛/=.q1/C2N . If k ! 1, we obtain

Trq,˛ Œu1  D u1 ,

thus u1 is self-similar. Therefore, for any r > 0, Trq,˛ Œu1 .1/ D u1 .1/ H) u1 .r / D r .2C˛/=.q1/ u1 .1/. Uniqueness of separable radial solutions of the equation implies u1 D Us . q,˛ r , thus Tr Œu  D u and If  ¤ RN , we denote uk :D u k k kr .2C˛/=.q1/C2N r .2C˛/=.q1/  r u1 .r x/ D u Trq,˛ Œu 1  D u1 H) r 1 .x/,

(4.2.22)

is an increasing function of , the same property for all r > 0 and x 2 r . Because u k r . If  is starshaped with respect to 0, r 7! u is shared by u 1 decreases. In the general 1 B0

R case there exist 0 < R < R0 such that BR    BR0 . Thus, uB  u  uk R k k

B

and u1r

1 R

B

r r  u 1  u1

1 R0

r . Let uQ :D limr !0 u 1 ; it is a solution of (4.2.1) in

121

Section 4.2 Isolated singularities

RN n ¹0º. From (4.2.22), q,˛

q,˛

r `r T` ŒTrq,˛ Œu 1  D T` Œu1  D u1

for ` > 0, therefore

(4.2.23)

T`q,˛ Œu Q D u, Q

and also 0 < uQ  Us . Then uQ is a non-zero separable solution of (4.2.1), under the form u.x/ Q D r .2C˛/=.q1/ !./, where ! > 0 solves q1

 !  N ,q,˛ ! C ! q D 0 on S N 1 . By the maximum principle ! N ,q,˛ , thus uQ D Us . We derive from (4.2.21) that the convergence 1=.q1/

1=.q1/ lim r .2C˛/=.q1/ u 1 .r x/ D N ,q,˛ ,

r !0

(4.2.24)

holds uniformly on ¹x : jxj D 1º. For proving uniqueness, we use again the fact that for any > 0, .1 C /u C is a supersolution of (4.2.1). It dominates near 0 any other solution u0 satisfying (4.2.18). If  ¤ RN (resp.  D RN ) it dominates u0 on @ (resp. if jxj ! 1). Thus u0  .1 C /u C H) u0  u. Similarly u  u0 .  The following result characterizes all the isolated singularities of the positive solutions of (4.2.1). Theorem 4.2.7. Assume that ˛ > 2, 1 < q < qe,˛ and  is an open domain of RN containing 0. If u 2 C 2 . n ¹0º/ is a positive solution of (4.2.1) in  n ¹0º, the following alternatives occur: (i) either lim jxj.2C˛/=.q1/ u.x/ D 1=.q1/ N ,q,˛ ,

r !0

(4.2.25)

(ii) or there exists k  0 such that u.x/ D k, r !0 N .x/

(4.2.26)

u C jxj˛ uq D kı0

(4.2.27)

lim

and u is a solution of

in the sense of distributions in . If k D 0, u is a bounded weak solution in . Proof. Without loss of generality we assume B2R  . It follows from (4.1.33), that d.x/ :D jxj˛ jujq1 .x/ is bounded from above in BR by CN0 ,q,˛ jxj2 . Let a be such that jaj D r  R=2. By the Harnack inequality in Br=4 .a/ and .4.2.3/ it follows that for any x with 0 < jxj  R=2, there holds max¹u.x/ : jxj D r º  C min¹u.x/ : jxj D r º,

(4.2.28)

122

Chapter 4 Isolated singularities

where C D C.N , q, ˛/ > 0. This isotropy is a strong characteristic of singularities of positive solutions. We define ˇ 2 Œ0, 1 by ˇ D lim sup x!0

u.x/ . N .x/

If ˇ D 0, (4.2.28) implies that u.x/ D o.N .x// near zero. By the maximum principle, u is bounded from above in BR n ¹0º by N C MR where M.R/ is the maximum of u on @BR and > 0 is arbitrary. This implies that u remains bounded, the singularity at 0 is removable and u is extended as an L1 weak solution of (4.2.1) in . If ˇ D 1, (4.2.28) implies u.x/ D 1, lim inf x!0 uk R is the solution of (4.2.1) in BR which vanishes on for any k > 0, where uk :D uB k @ and satisfies (4.2.11). By the maximum principle u  uk and thus u  u1 . Hence by Theorem 4.2.6 1=.q1/ (4.2.29) lim inf jxj.2C˛/=.q1/ u.x/  N ,q,˛ .

x!0

As in the proof of Corollary 4.2.2, for any ı 2 .0, R/, u is bounded from above in BR n Bı by the solution w :D wı ,MR of w C jxj˛ w q D 0 wD1 w D MR

in BR n Bı on @Bı on @BR .

(4.2.30)

Notice that w is radial and satisfies the differential equation w 00 

N 1 0 w C r ˛ w q D 0 in .ı, R/. r

(4.2.31)

For existence we construct the solutions wk :D wk,ı ,MR of (4.2.30) such that wk,ı ,MR .ı/ D k (k > 0). Since wk is increasing with respect to k and is bounded in Œı 0 , R for any ı 0 2 .ı, R/ by Keller–Osserman, existence of wı ,MR follows by letting k ! 1. Using the fact that ı 7! wı ,MR is increasing, we obtain u  w0,MR :D limı !0 wı ,MR by letting ı ! 0. Similarly Us  w0,MR0 where .2C˛/=.q1/ . Moreover, it follows from the construction of w MR0 D 1=.q1/ ı ,MR N ,q,˛ R and wı ,MR0 , which are the limits (as n ! 1) of solutions with data n on @Bı , that

jw0,MR  w0,MR0 j  jMR  MR0 j.

(4.2.32)

Furthermore, using (4.2.21), we have for r > 0, 0 0 Trq,˛ Œwı ,MR0  D wı=r ,MR=r H) Trq,˛ Œw0,MR0  D w0,MR=r .

(4.2.33)

123

Section 4.2 Isolated singularities

If we let R ! 1, then limR!1 w0,MR0 D w  is self-similar. Hence it coincides with Us by the argument used in Theorem 4.2.6 in the case  D RN . When r ! 0 in (4.2.33), there holds lim r .2C˛/=.q1/w0,MR0 .r / D Us .1/ D N ,q,˛ . 1=.q1/

r !0

(4.2.34)

Combining (4.2.34), (4.2.32), (4.2.29) with the fact that u  w0,MR , we obtain statement (4.2.25). If 0 < ˇ < 1 there holds from (4.2.28) lim inf x!0

u.x/ D k  ˇ=C . N .x/

This implies u.x/ D 1, x!0 uk .x/ where uk is the solution of (4.2.1) in BR n ¹0º, which vanishes on @BR and satisfies (4.2.11). For any k 0 < k, there exists a punctured neighborhood of 0 in which u.x/ > uk 0 .x/. Thus u > uk 0 in BR n ¹0º and finally u  uk ; actually the inequality is strict by the strong maximum principle. Moreover there exists a sequence ¹xn º converging to 0 such that u.xn / D 1 ” u.xn /  uk .xn / D o.N .xn //. lim n!1 uk .xn / lim inf

Set w D u  uk , then w C d.x/w D 0 w>0 w>0 where d.x/ D

in BR n ¹0º in BR n ¹0º near @BR ,

8 < jxj˛ uq uqk

if u > uk

:0

if u D uk .

uuk

(4.2.35)

(4.2.36)

Thus 0  d.x/  C jxj2 by (4.1.33), and by the Harnack inequality (see Theorem 1.6.1), max w.x/  C

jxjDjxnj

min w.x/ D o.N .xn / D o.N .jxn j/.

jxjDjxnj

(4.2.37)

For > 0, the function w D N CMR satisfies w Cd.x/w  0 in BR nB jxn j . Therefore it dominates w in this domain. If we let successively xn ! 0 and ! 0, we obtain 0  u  uk  MR in BR n ¹0º. (4.2.38) We prove that u satisfies (4.2.27) in the same way as in Theorem 4.2.5 (ii). Clearly if k D 0, u is a bounded weak solution. 

124

Chapter 4 Isolated singularities

4.2.3 Isolated signed singularities We recall that .r , / denote the spherical coordinates in RN and that solutions of .4.2.1/ of the form 2C˛ u.r , / D r  q1 !./ exist if ! satisfies  !  N ,q,˛ ! C j!jq1 ! D 0 in S N 1

(4.2.39)

where N ,q,˛ is defined by (4.2.2). We denote by EQ S N 1 the set of ! 2 C 2.S N 1 / solutions satisfying (4.2.39). q,˛

Proposition 4.2.8. Assume that N  2, ˛ > 2 and q > 1. Then (i) If q  qe,˛ :D

N C˛ , N 2

EQ Sq,˛ N 1 D ¹0º.

± ° 1 2 .S N 1 / D 0,  q1 (ii) If 1 < q < qe,˛ , EQ Sq,˛ \ C N 1 C N ,q,˛ . (iii) If

N C1C˛ N 1

± ° 1 q,˛ q1 :D qc,˛  q < qe,˛ , EQ S N 1 D 0, ˙N ,q,˛ .

(iv) 1 < q < qc,˛ , there exist signed elements in EQ S N 1 . q,˛

Proof. If † is a domain of S N 1 and ! 2 W01,2.†/ \ LqC1.†/, we denote  Z  1 2 0 2 2 qC1 jwj jr !j  N ,q,˛ ! C dS./. (4.2.40) J† .!/ D 2 † qC1 Then

J†0 .!/ D  !  N ,q,˛ ! C j!jq1 ! on S N 1 . q,˛ If we assume that ! 2 EQ N 1 has maximum !max and minimum !min , then S

N ,q,˛ !max  j!max jq1 !max and N ,q,˛ !min  j!min jq1 !min . If q  qe,˛ , N ,q,˛ is non-positive; the previous inequalities imply !max  0  !min and (i) follows. If 1 < q < qe,˛ , N ,q,˛ > 0, thus EQ Sq,˛ N 1 contains at least the three 1

q1 constants 0 and ˙N ,q,˛ . If we assume that ! is a positive non-constant element of 1

q,˛ q1 EQ S N 1 and set ` D !max =N ,q,˛ , then `  1 and `! is a positive subsolution of 1

q1 (4.2.39). The function w D N ,q,˛  `! vanishes at some 0 . It is non-negative, non-constant and satisfies

 w  N ,q,˛ w C dw  0 where d is non-negative. The contradiction follows by the strong maximum principle; thus (ii) holds.

125

Section 4.2 Isolated singularities

We assume now that qc,˛  q < qe,˛ . Let N be the spherical average on S N 1 of

2 L1.S N 1 /. If 2 L2.S N 1 /, N is the orthogonal projection onto the kernel of  . From (4.2.39), we get Z   jr 0 .!  !/j N 2  N ,q,˛ .!  !/ N 2 C .j!jq1 !  j!jq1 !/.!  !/ N dS./ D 0. S N 1

The first non-zero eigenvalue of  is N 1. Since !  !N is orthogonal to the kernel of  , we have Z  .!  !/ N  .!  !/dS./ N S N 1 Z Z 0 2 D jr .!  !/j N dS./  .N  1/ .!  !/ N 2 dS./. S N 1

S N 1

Furthermore Z .j!jq1 !  j!jq1 !/.!  !/dS./ N N 1 S Z D .j!jq1 !  j!j N q1 !/.! N  !/dS./. N S N 1

Since qc,˛  q H) N  1  N ,q,˛ and .j!jq1 !  j!j N q1 !/.! N  !/ N  21q j!  !j N qC1 , the series of above inequalities imply !  !N D 0, which proves (iii). Finally, if † is a subdomain of S N 1 and † the first eigenvalue of  in W01,2.†/, for any  > † we can construct a positive solution ! 2 W01,2 .†/ \ LqC1.†/ to  !  ! C j!jq1 ! D 0 in †,

(4.2.41)

by minimizing the functional J†,

 Z  1 2 0 2 2 qC1 jr !j  ! C dS./. jwj J†, .!/ D 2 † q C1 Any solution ! of .4.2.41/ satisfies 1=.q1/  !  1=.q1/ by the maximum principle and is C 2 in †. If † is a Lipschitz spherical domain, ! is continuous in †. This is the case if † is the fundamental simplicial domain of a finite subgroup G of the orthogonal group O.N / generated by reflections through hyperplanes, then @† is a finite union of pieces of unit spheres of dimension N  2 on S N 1 (great circles in N D 3). If ! is a positive solution of .4.2.41/ vanishing on @†, we can extend it by reflection through @† into a signed solution !† of the same equation in whole S N 1 . Such a construction, which uses a tessellation of the sphere, is possible in particular if N 1 . This ends the proof of (iv).  1 < q < qc,˛ with † D SC

126

Chapter 4 Isolated singularities

Remark 4.4. The fine structure of the set EQ S N 1 is not known except if qc,˛  q < qe,˛ (see (iii)) or if N D 2. In that case ± ° q,˛ (4.2.42) EQ S 1 :D ! 2 C 2 .S 1 / : !  2,q,˛ ! C j!jq1 ! D 0 in S 1 . q,˛

For such an equation, standard quadrature techniques lead to a full description of the 1=.q1/ set EQ q,S 1 : it consists in the three constant solutions ¹0, ˙2,q,˛ º and the union of connected components generated by  7! !j . Cˇ/, where !j is a solution pwith least anti-period =j where j ranges from 1 to the largest integer smaller than 2,q,˛ . The following result gives a general description of the behavior of solutions of .4.2.1/ near 0. Theorem 4.2.9. Assume that ˛ > 2, q > 1 and  is an open subset of RN containing 0. If u 2 C 2 . n ¹0º/ is a solution of (4.2.1), then r .2C˛/=.q1/u.r , ./ converges q,˛ as r ! 0 to a connected and compact subset of EQ S N 1 in the C 2 -topology. The limit is zero if q  qe,˛ . Proof. By convergence toward a set, we mean that there exists a connected and compact subset E 0 of EQ Sq,˛ N 1 such that lim dist C 2 .r .2C˛/=.q1/ u.r , ./, E 0 / D 0,

r !0

(4.2.43)

where dist C 2 . , E 0/ D inf¹k  !kC 2 .S N 1/ : ! 2 E 0 º. We assume that B 2R  . By (4.2.3), r .2C˛/=.q1/u.r , / is bounded on .0, R S N 1 . If we set v.t , / D r .2C˛/=.q1/u.r , / with t D ln r , then   qC1C˛ v t C N ,q,˛ v C  v  jvjq1 v D 0 (4.2.44) vt t C N  2 q1 in .1, ln R/ S N 1 . By standard regularity estimates, @ˇ r˛0 v=@t ˇ are bounded in .1, ln R  1/ S N 1 for 0  ˇ C j˛j  3, where r˛0 denotes S the covariant derivatives to the order j˛j. Thus the negative trajectory T Œv D t
It is a non-empty compact and connected set and there holds lim dist C 2 .v.t , ./, E 0 / D 0.

t !1

We define the energy function by   Z 1 2 0 2 2 qC1 2 EŒv D jr vj  v t C  N ,q,˛ v dS./. jvj 2 S N 1 qC1

(4.2.46)

(4.2.47)

127

Section 4.2 Isolated singularities

Multiplying (4.2.44) by v t and integrating on S N 1 yields  Z d q C1C˛ EŒv D N  2 v 2t dS./. dt q1 S N 1

(4.2.48)

The coefficient of v t , called the damping coefficient, is not zero (it vanishes only if q D .N C 2 C 2˛/=.N  2/ > qe,˛ ). Thus for any T < ln R0  ln R  1   Z 0 Z q C 1 C ˛ 1 ln R N 2 v 2t dS./dt D EŒv.ln R/EŒv.T /. (4.2.49) q1 2 T S N 1 Since the a priori estimates imply that EŒv is bounded on .1, ln R0 , we obtain Z

ln R0 1

Z S N 1

v 2t dS./dt < 1.

Differentiating the equation with respect to t and setting w D v t , we get   q C1C˛ w t C N ,q,˛ w C  w  qjvjq1 w D 0. wt t C N  2 q1

(4.2.50)

(4.2.51)

Define 1 E Œw D 2 

Z

 S N 1

 jr 0 wj2  w 2t  N ,q,˛ w 2 dS./,

(4.2.52)

then Z  Z d  qC1C˛ E Œw D N  2 w 2t dS./  q ww t jvjq1 dS./. N 1 N 1 dt q1 S S (4.2.53) Again E  Œw is bounded on .1, ln R0 . Using (4.2.50) and the Cauchy–Schwarz inequality, Z ln R Z v 2t t dS./dt < 1. (4.2.54) 1

S N 1

Inequalities (4.2.50) and (4.2.54), combined with the uniform continuity of t 7! v t .., t / and t 7! v t t .., t / on .1, ln R0  (which are a consequence of the a priori estimates) imply lim jjv t .t , ./jjL2 .S N 1 / D lim jjv t t .t , ./jjL2 .S N 1/ D 0.

t !1

t !1

(4.2.55)

Since any element ! of E 0 is the limit of v¹.tn, ./º for some sequence ¹tnº converging q,˛  to 1, (4.2.44) implies that ! belongs to EQ S N 1 .

128

Chapter 4 Isolated singularities

Remark 4.5. It is an interesting and deep question to know whether E 0 is reduced to a single element or not. This is always true if qc,˛  q < qe,˛ since EQ Sq,˛ N 1 is discrete, a result which is no longer valid if 1 < q < qc,˛ since all but three connected components of this set are continuum. However, if N D 2 and ˛ D 0, it is proved in [30] that the limit set is a singleton. The method relies essentially on Sturm intersection curve arguments and it can be extended to the case ˛ > 2. Another important question is to know what happens when E 0 D ¹0º. The next result, of which we give an abridged proof below, shows that u behaves like a harmonic function, may be singular at ¹0º, with a singularity weaker than r .2C˛/=.q1/. In order to analyze the linearized problem we recall that the spectrum . / of  in W 1,2.S N 1 / is the set of the k D k.N C k  2/, k D 0, 1, ... Theorem 4.2.10. Assume  is as in Theorem 4.2.9, ˛ > 2 and 1 < q < qe,˛ . If u 2 C 2 . n ¹0º/ is a solution of (4.2.1) satisfying limr !0 r .2C˛/=.q1/u.r , ./ D 0, there exist an integer k 2 Œ0, 2qC˛ q1  N / and a non-zero spherical harmonic k of degree k such that lim r N Ck2u.r , ./ D k , (4.2.56) r !0

(with the usual modification if N D 2) if one of the following conditions is satisfied: (i) N=2. (ii) qc,˛  q < qe,˛ . (iii) N  3 and N ,q,˛ … . bS N 1 / D ¹`.N C `  2/ : ` 2 Nº. Proof. If N ,q,˛ … . / we use the logarithmic variable and the function v as in the proof of Theorem 4.2.9, the main point is to prove that there exist some C > 0 and 0 > 0 such that jjv.t , ./jjL1 .S N 1/  C e 0t , (4.2.57) for t  ln R  1. If we assume that this decay does not hold, there exists a function  2 C 1 .1, ln R with the following properties (i)

 > 0, 0 > 0, lim t !1 .t / D 0

(ii) 0 < lim supt !1 1 .t /jjv.t , ./jjL1 .S N 1/ < 1 (iii) lim t !1 e t D 1, for all > 0 (iv) .0 =/0, .00 =/0 2 L1.1, ln R/ (v) lim t !1 0 .t /=.t / D lim t !1 00 .t /=.t / D 0. The construction of  is made by approximating from below the function t 7! jjv.t , ./jjL1 by a continuous piecewise linear function and then by smoothing it. The

129

Section 4.2 Isolated singularities

detailed construction is given in [30]. Furthermore vQ D 1 v satisfies   qC1C˛ 0 vQ t t C N  2 C2 vQ t q1    0 00 C N ,q,˛ C a C Q q1 vQ D 0 (4.2.58) vQ C  vQ  q1jvj   in .1 ln R/ S N 1 , where a D N  2.q C 1 C ˛/=.q  1/. The functions 0 = and 00 = are small perturbations because of property (v), therefore vQ and all its derivatives up to the order 3 remain bounded. Therefore, the limit set EQ 0 at 1 of the trajectory of v, Q defined similarly to (4.2.45), is non-empty, connected and compact. Property (iv) allows to use the energy method as in Theorem 4.2.9; while properties (ii) and (v) imply that EQ 0 is a nontrivial subset of the set of solutions of  C N ,q,˛ D 0 N 1

. This is a contradiction since N ,q,˛ is not an eigenvalue. Thus (4.2.57) holds. on S The remainder of the proof is a more standard linearization process which is carried out in studying the equation satisfied by w0 :D e 0 t v, by projection onto the eigenspaces of  and using the energy method as before. It follows that the limit set E00 of the negative trajectory TŒw0  is a subset of the set of solutions of  C .N ,q,˛  0 / D 0. If N ,q,˛  0 2 . / and E00 ¤ ¹0º, the projection of the equation satisfied by w0 onto the eigenspace associated to N ,q,˛  0 , which is a finite dimensional system, shows that E00 is reduced to a nontrivial and unique element. This yields statement (4.2.56). If E00 D ¹0º, we project again the equation onto the eigenspaces of  and derive that the decay of v is actually stronger since there holds jjv.t , ./jjL1  C e 20t .

(4.2.59)

Iterating this bootstrap procedure, we finally obtain its stabilization in a finite number of steps. This implies that u satisfies (4.2.56). If qc,˛  q < qe,˛ , and in particular if q D qc,˛ , otherwise the problem is nonspectral and treated by the above method, the argument used to study the structure of Eq,S N 1 applies and yields N /jjL1 .S N 1 / D 0. lim r N 1jju.r , ./  u.r

r !0

From this isotropy result, it follows by arguing by contradiction and using the maximum principle that u must have a constant sign near 0. Then the proof follows from Theorem 4.2.7. 1=2 Notice that, if N D 2, the critical spectral case 2,q,˛ 2 N is solved by a delicate adaptation of Sturm intersection curve arguments combined with a refined version of the -function already used in the non-spectral case. This argument is developed in [30]. 

130

Chapter 4 Isolated singularities

4.3 Boundary singularities 4.3.1 Upper bounds The following result is the analog of Theorem 4.1.2. Theorem 4.3.1. Assume   RN is a domain such that 0 2 @ and g 2 GQe ./ satisfies the Keller–Osserman condition. There exists a non-decreasing continuous function f : RC 7! R which verifies lim!0 f ./ D 1 and lim!1 f ./ D 1 such that if u 2 C.n¹0º/\C 2./ coincides on @n¹0º with a continuous function 2 C.@/ and satisfies u C g ı u  0 (4.3.1) in , then u.x/  f .jxj/ C max .max¹ .x/ : x 2 @ \ Br º; max¹u.x/ : x 2  \ @Br º/ , (4.3.2) for all x 2  \ Br and r > 0. Proof. Without loss of generality, we can assume that  is bounded. Thus there exists m  0 such that u  m < 0 on @ n ¹0º. Since g ı u  h ı u  h.u  m/, there holds by Kato’s inequality .u  m/C C h..u  m/C /  0.

(4.3.3)

Since .u  m/C is continuous in  n ¹0º and vanishes in a neighborhood of @ n ¹0º, its extension w by zero outside  satisfies (4.3.3) in RN n ¹0º, in the weak L1-sense. Therefore w.x/  f .jxj/ for x ¤ 0 by Theorem 4.1.2. This implies (4.3.2).  Similarly, Theorem 4.1.4 and Theorem 4.1.5 admit extensions to the boundary singularity problem under the assumption that g 2 GQ0./ and satisfies the corresponding Keller–Osserman condition. Corollary 4.3.2. Let  be as in Theorem 4.3.1, ˛ > 2, q > 1 and u 2 C. n ¹0º/ \ C 2./ be a solution of (4.2.1) in  which coincides on @ n ¹0º with a bounded function 2 C.@/. Then ju.x/j  CN ,q,˛ jxj.2C˛/=.q1/ C jj jjL1 .@/ .

(4.3.4)

If we assume moreover that @ is bounded and C 2 and D 0, then ju.x/j  C.x/jxj.qC1C˛/=.q1/, where C D CN ,q, > 0.

(4.3.5)

131

Section 4.3 Boundary singularities

Proof. Estimate (4.3.4) follows from (4.1.33) and the previous theorem. For the second assertion there exists r0 > 0 such that @ \ Br0 is a C 2 graph. Let G be a C 2 domain such that @G \ B5 is the graph of some C 2 function . If v is any solution of (4.2.1) in G \ B5 vanishing continuously on @G \ B5 n ¹0º, it is uniformly bounded in G1,4 :D G \ .B4 n B 1 /. Then rv is bounded in G2,3 :D G \ .B3 n B 2/ by some constant depending on N , q and the curvature of @G \ B5, which is controlled by the C 2 norm of (see [53, Ch. 3]). Therefore, if 0 < jyj < 5, there exists r 2 .0, r0 =5/ q,˛ such that 0 < r jyj  r0. If v D Tr Œu, defined by (4.2.21), then v satisfies (4.2.1) in r \ B5 (we recall that r D r 1 ) and is bounded in r \ .B4 n B 1 /. Since the upper bound on r implies the uniform boundedness of the curvature of @r \ B5, rv is bounded in r \ .B3 n B 2 / by a constant independent of r . Therefore, for all ry D x 2  \ .B3r n B 2r /, jru.x/j  C r .qC1C˛/=.q1/  3.qC1C˛/=.q1/ C jxj.qC1C˛/=.q1/ .

(4.3.6)

This estimates holds for any r 2 .0, r0=5/ or equivalently any x 2  \ Br0 . Because u vanishes on @ n ¹0º, (4.3.6) implies (4.3.5).  Remark 4.6. If u belongs to C. n ¹0º/ and satisfies u C jxj˛ jujq1 u  0

(4.3.7)

in the weak L1-sense, then (4.3.4) is replaced by a one-sided estimate u.x/  CN ,q,˛ jxj.2C˛/=.q1/ C jj C jjL1 .@/ .

(4.3.8)

4.3.2 The half-space case 0 N We shall denote x D .x1, ..., xN / the coordinates in RN , RN C D ¹x D .x1 , x / 2 R : C N N N x1 > 0º, @RC :D ¹x 2 R : x1 D 0º and BR D BR \ RC . The upper hemisphere N 1 N 1 of the unit sphere is SC :D S N 1 \ RN  S N 2 . C , with relative boundary @SC N N There exist solutions of (4.2.1) in RN C , continuous in RC n ¹0º and vanishing on @RC n ¹0º under the form (4.3.9) u.r , / D r .2C˛/=.q1/ !./,

if ! is a non-zero solution of the problem  !  N ,q,˛ ! C j!jq1 ! D 0 !D0

N 1 in SC , N 1 . on @SC

(4.3.10)

q,˛ We denote by EQ N 1 the set of solutions of (4.3.10). As an extension of ProposiSC

tion 4.2.8, we have

132

Chapter 4 Isolated singularities

Proposition 4.3.3. (i) If q  qc,˛ :D

N C1C˛ N 1 ,

q,˛ EQ N 1 D ¹0º. SC

Q q,˛ , therefore (ii) If 1 < q < qc,˛ , EQ q,˛ N 1 admits a unique positive element ! :D ! SC

q,˛ N 1 EQ N 1 \ CC .SC / D ¹0, !Q q,˛ º. SC

(iii) If qi ,˛ :D

N C2C˛ N

(iv) If 1 < q < qi ,˛ , EQ

q,˛  q < qc,˛ , EQ N 1 D ¹!Q q,˛ , !Q q,˛ , 0º. SC

q,˛ N 1 SC

contains signed elements besides 0 and ˙!Q q,˛ .

Proof. Statement (i) follows by multiplying the equation by ! and using the fact that q  qc,˛ implies N ,q,˛  N  1 which is the first eigenvalue of  in N 1 /. If 1 < q < qc,˛ , there exists a nontrivial non-negative minimizer W01,2 .SC 1,2 N 1 N 1 / \ LqC1.SC / to the functional !Q q,˛ 2 W0 .SC Z   1 2 j!jqC1 dS./. jr 0 !j2  N ,q,˛ ! 2 C ! 7! JQS N 1 .!/ :D C N 1 2 SC qC1 By the strong maximum principle and standard regularity theory, !Q q,˛ is positive in q,˛ N 1 N 1 and belongs to C 2.SC /. If ! and ! 0 are two positive elements of EQ N 1 , SC Z

SC

  ! 0  ! q1 0 q1 C .! 2  ! 02 /dS./ C j!j  j! j N 1 ! !0 SC ˇ ˇ Z ˇ ! 0 0 ˇˇ2 ˇˇ 0 0 ! 0 0 ˇˇ2 ˇ 0 D ˇr !  0 r ! ˇ C ˇr !  r ! ˇ N 1 ! ! SC  C .j!jq1  j! 0 jq1 /.! 2  ! 02 / dS./.

0D





Integration by part is justified by Hopf’s lemma. Thus ! D ! 0 and this proves (ii). N 1 . For proving (iii) we explicate the representation of SC °  ± N 1 (4.3.11) D x D .cos , sin  0 / :  0 2 S N 2 , 2 Œ0, / . SC 2 Then  ! D



1 N 2

sin



sinN 2 !

 

C

1 sin2

0 !

where 0 is the Laplace–Beltrami operator on S N 2 . Furthermore, the volume element measure on S N 1 induced by the imbedding of S N 1 into RN is expressed by dS./ D sinN 2 dS 0 . 0 /d where dS 0 is the volume element on S N 2 . If N 1  2 L1.SC /, we denote Z 1 . / :D N 2 .., /dS 0 . jS j S N 2

133

Section 4.3 Boundary singularities

The first eigenfunction of  is ! 7 cos , with corresponding eigenvalue N  1, and  Z Z Z 2 0 .  / cos dS./ D .  / dS cos sinN 2 d D 0. N 1 SC

S N 2

0

N 1 /. Therefore    has no component in the first eigenspace of  in W01,2 .SC Since the second eigenvalue is 2N , it follows Z Z  .  / .  /dS./  2N .  /2 dS./. N 1 SC

N 1 SC

This holds in particular if  D !. Furthermore Z .!  !/.j!jq1 !  j!jq1 !/dS./ N 1 SC

Z D

N 1 SC

.!  !/.j!jq1 !  j!jq1 !/dS./ Z C

Since Z N 1 SC

N 1 SC

.!  !/.j!jq1 !  j!jq1 !/dS./.

.!  !/.j!jq1 !  j!jq1 !/dS./ Z

2

D 0

Z S N 2

 .!  !/ dS 0 .j!jq1 !  j!jq1 !/ sinN 2 d

D 0, it follows Z N 1 SC

.!  !/.j!jq1 !  j!jq1 !/dS./  21q

Z N 1 SC

j!  !jqC1 dS./.

If q  qi ,˛ , N ,q,˛  2N , thus !  ! D 0 which shows that ! depends only on the

variable and it satisfies ! 00  .N  2/ cot ! 0  N ,q,˛ ! C j!jq1 ! D 0 on .0, 2 / ! 0 .0/ D 0, !. 2 / D 0.

(4.3.12)

If we assume !.0/ > 0 and ! changes signs at some 0 2 .0, 2 /, then we apply Courant’s formula to the spherical caps ¹ : 0 < < 0 º and ¹ : 0 < < 2 º N 1 which are domains on SC : ³ ² Z 0 Z 2 ! 02 sin N 2 d , ! 02 sin N 2 d . 2N  max 0

0

134

Chapter 4 Isolated singularities

Therefore, ²Z

0

min

j!jqC1 sin N 2 d ,

0

Z

2

0

³ j!jqC1 sin N 2 d  0,

which implies that ! vanishes either on .0, 0 / or . 0 , 2 /, a contradiction. Thus any q,˛ solution keeps a constant sign and EQ N 1 is reduced to 0 and ˙!Q q,˛ . SC

q,˛ For proving (iv), if 1 < q < qi ,˛ , one can construct signed elements of EQ N 1 by conSC

sidering a quarter sphere as a fundamental domain of the group of reflections generated by the symmetries with respect to the planes x1 D 0 and x2 D 0. The corresponding N 1 of the eigenvalue is 2N and if q < qi ,˛ , then N ,q,˛ > 2N . The restriction to SC q,˛ Q solution constructed in Proposition 4.2.8 (iv) is a signed element of E N 1 .  SC

The corresponding solution of (4.2.1) is denoted by UQ 0 .x/ :D jxj.2C˛/=.q1/ !Q q,˛ .x=jxj/ D r .2C˛/=.q1/!s ./.

(4.3.13)

The following result dealing with isolated singularities of a positive solution has been proved in [54] in the case ˛ D 0. It that case it is also a consequence of Theorem 3.4.10, although the method used here extends easily to .4.2.1/. In this chapter it is a consequence of a more general result dealing with signed solutions. Theorem 4.3.4. Assume N  2, 1 < q < qc,˛ and u is a positive solution of .4.2.1/ C C C :D BR \ RN in BR C which is continuous in BR n ¹0º and vanishes on @BR n ¹0º. Then: (i) either u.x/ D1 UQ 0 .x/

(4.3.14)

u.x/ Dk x!0 P .x, 0/

(4.3.15)

lim

x!0 N 1 , uniformly on SC (ii) or there exists k  0 such that

lim

N 1 uniformly on SC , where P .x, y/ is the Poisson kernel in RN C.

For signed solutions, a straightforward adaptation of Theorem 4.2.9 yields: C C n ¹0º/ \ C 2 .BR / is a solution Theorem 4.3.5. Assume ˛ > 2, q > 1. If u 2 C.BR N .2C˛/=.q1/ u.r , ./ converges as r ! 0 of (4.2.1) vanishing on .BR \@RC /n¹0º, then r 2 to a connected and compact subset EQ 0 of EQ q,˛ N 1 in the C -topology. In particular, if

q  qc,˛ , then EQ 0 D ¹0º.

SC

135

Section 4.3 Boundary singularities

Remark 4.7. As in the proof of Theorem 4.2.9, the energy function by   Z 2 1 Q jvjqC1  N ,q,˛ v 2 dS./, jr 0 vj2  v 2t C EŒv D N 1 2 SC qC1 plays a fundamental role since  Z  qC1C˛ 1 d Q v 2t dS./. EŒv D N  2 N 1 dt q1 2 SC

(4.3.16)

(4.3.17)

Q Therefore t 7! EŒv.t / is monotone. In the same way as in the proof of Theorem 4.2.9, q,˛ there exists ! 2 EQ N 1 such that SC

lim rn.2C˛/=.q1/ u.rn , ./ D !,

rn !0

for some sequence ¹rn º converging to 0. The limit set at 1 of the negative trajectory of v is defined by N 1 C 2 .SC / \ [ 0 Q ¹v., ./º . (4.3.18) E D t
If ! 2 EQ 0 there holds

Q /. JQS N 1 .!/ D lim EŒv.t C

t !1 Q0

(4.3.19)

In other words, the energy is constant on E . We denote its value by JQS N 1 .EQ 0 /. C q,˛ Finally, the energy on EQ N 1 can be easily computed by using equation (4.3.10): SC

JQS N 1 .!/ D C

1q 2.q C 1/

Z

N 1 SC

j!jqC1 dS./.

(4.3.20)

Similarly to Theorem 4.2.10, one obtains: Theorem 4.3.6. Let the assumptions of Theorem 4.3.5 be fulfilled with in addition 1 < q < qc,˛ and assume limr !0 r .2C˛/=.q1/u.r , ./ D 0. Then there exist an integer k 2 Œ1, 2qC˛ q1  N / and a non-zero spherical harmonic of degree k, vanishing on N 1 , such that @SC (4.3.21) lim r N Ck2u.r , ./ D k , r !0

if one of the following conditions is satisfied: (i) N=2. (ii) qi ,˛  q < qc,˛ . (iii) N  3 and N ,q,˛ … . bS N 1 / D ¹`.N C `  2/ : ` 2 N º. C The positive solutions defined in the complement of a RN C n BR have only one type of decay at infinity.

136

Chapter 4 Isolated singularities

Theorem 4.3.7. Assume N  2, 1 < q < qc,˛ and u is a positive solution of .4.2.1/ Cc C Cc Cc N :D RN in BR C n BR which is continuous in BR and vanishes on BR \ @RC . Then u.x/ D1 Q 0 .x/ jxj!1 U lim

(4.3.22)

N 1 uniformly on SC . 2C˛

Proof. By .4.3.4/, u.x/  CN ,q,˛ jxj q1 . Up to changing R we can assume that Cc C / therefore ru is bounded from above on @BR and there exists a > 1 u 2 C 2 .BR @u Q such that u.x/  aU0 .x/ for x D R. By Hopf’s lemma, @x > 0 for x D R and 1 x1 D 0. Since u > 0 for x D R and x1 > 0 by the strong maximum principle, there exists b 2 .0, 1/ such that u.x/  b UQ 0 .x/ for x D R. Finally, since aUQ 0 and b UQ 0 are respectively a super- and a subsolution of .4.2.1/, it follows that

b UQ 0 .x/  u.x/  aUQ 0 .x/

Cc 8x 2 BR .

(4.3.23)

For r > 0, set ur .x/ D Trq,˛ Œu.x/, then ur satisfies .4.2.1/ in BrC1cR and .4.3.23/ holds. By compactness there exist a sequence ¹rnº tending to 1 and v 2 C.RN C n ¹0º/ 1 N vanishing on @RN C n ¹0º such that urn ! v in the Cloc topology of RC n ¹0º and

b UQ 0 .x/  v.x/  aUQ 0 .x/

8x 2 RN C n ¹0º.

(4.3.24)

1 By Theorem 3.4.6, v D UQ 0 and by uniqueness ur ! UQ 0 in the Cloc topology as r ! 1. Therefore 2C˛

lim r q1 u.x/ D !s .

jxj!1

x / and jxj

2C˛

lim r q1 r 0 u.x/ D r 0 !s .

jxj!1

x /, jxj 

which implies .4.3.22/.

A simple adaptation of Theorem 4.3.5 yields the following description of the asymptotic behavior of signed solutions. Theorem 4.3.8. Assume N  2, ˛ > 2 and q > 1. If u is a solution of .4.2.1/ in Cc C Cc Cc N BR :D RN C n BR which is continuous in BR and vanishes on BR \ @RC , then .2C˛/=.q1/ r u.r , ./ converges as r ! 1 to a connected and compact subset EQ 1 of q,˛ EQ N 1 in the C 2-topology. SC

2C˛

Set v.t , ./ D r q1 u.r , ./ with t D ln r , then Sthis theorem characterizes the limit set at infinity of the positive trajectory TCŒv D t >0 ¹v.t , ./º, which is defined by EQ 1 D

\ [ t >ln R >t

¹v., ./º

N 1 C 2 .SC /

.

137

Section 4.3 Boundary singularities

Similarly as in Theorem 4.3.5 and the remark hereafter, the energy function JQS N 1 is C constant on EQ 1 , with value denoted by JQ N 1 .EQ 1 /. SC

In order to study the global solution, we associate a semigroup (or semiflow) of continuous operators to equation (4.2.1). We denote by B1C the intersection of B1 c

Cc C N with RN the set RN C , by B1 C \ B 1 , by @ B1 the intersection of @B1 with RC and by c N c @ B1 the set @RC \ B1 .

Lemma 4.3.9. Assume q > 1, ˛ > 2. Then for any 2 C.@CB1 / there exists a unique u 2 C.B1C c / \ C 2.B1C c / satisfying u C jxj˛ jujq1 u D 0 uD

uD0

in B1C c on @C B1 on @c B1 .

(4.3.25)

Furthermore if we set 2C˛

S t Π./ D r q1 u.r , /

N 1 with t D ln r > 0 and  2 SC ,

(4.3.26)

N 1 N 1 / and for any 2 C.SC /, then for t > 0, S t is a compact mapping in C.SC N 1 t 7! S t Œ  is continuous from RC into C.SC /. Furthermore the semigroup property holds: S t Cs D S t ı Ss for s, t  0 and S0 D I .

Proof. Existence follows by standard approximation arguments and uniqueness by the maximum principle and the fact that the following estimate holds   2C˛ (4.3.27) ju.r , /j  min k kL1 , CN ,q r  q1 . Clearly, uniqueness implies the semigroup property. The compactness and the continuity follow from standard regularity estimates.  N To any u 2 C.RN C n¹0º/, solution of (4.2.1) vanishing on @RC n¹0º, we can associate

N 1 // with a global orbit of the semiflow ¹S t º t 0, that is a function v 2 C.R; C.SC the property that v.t C s/ D S t Œv.s/ 8s 2 R , t  0. (4.3.28) S A global orbit T Œv :D t 2R ¹v.t , ./º has the property that its limit sets at infinity EQ 1 and at minus infinity EQ 0 are not empty and Z 1 Z  qC1C˛ v 2t dS./dt . JQS N 1 .EQ 1 /  JQS N 1 .EQ 0 / D N  2 C C N 1 q1 1 SC (4.3.29)

138

Chapter 4 Isolated singularities

Proposition 4.3.10. Assume q > 1 and ˛ > 2. Let u 2 C.RN C n ¹0º/, solution of N 0 1 Q Q (4.2.1) vanishing on @RC n ¹0º and let E and E be defined respectively in Theorem 4.3.5 and Theorem 4.3.8. Then JQS N 1 .EQ 1 /  JQS N 1 .EQ 0 /. If JQS N 1 .EQ 1 / D C

C

C

q,˛ JS N 1 .EQ 0 /, then there exists some ! 2 EQ N 1 such that u.r , / D r  q1 !./. 2C˛

SC

C

N 1 and there exists Furthermore, if !0 , !1 2 EQ q,˛ N 1 are such that !0 < !1 in SC SC

q,˛ no other ! 2 EQ N 1 such that !0  !  !1 , there exists a global orbit v such that SC

limt !1 v.t , ./ D !1 and lim t !1 v.t , ./ D !0 . In particular, !1 can be the unique q,˛ positive element !Q q,˛ of EQ N 1 and !0 any other element. SC

Proof. From (4.3.29), a necessary condition for existence on a heteroclinic global orbit, i.e., a global orbit v connecting two different elements !0 D lim tn !1 v.t , ./ q,˛ and !1 D lim tn !1 v.t , ./ of E N 1 , is SC

JQS N 1 .!1 / < JQS N 1 .!0 /, C

C

(4.3.30)

while, if JQS N 1 .!1 / D JQS N 1 .!0 /, the same identity implies that v t D 0 and thus v is C C constant. Furthermore, existence of connecting orbits between two stationary solutions is proved in [87, Theorem 8], [88], under a strict order condition as in the statement of the proposition and the non-existence of any other stationary solutions in the order interval Œ!0 , !1 .  Remark 4.8. It is a deep question to know if the condition (4.3.30) is sufficient for the existence of a heteroclinic global orbit between them. In dimension N=2, it is proved in [89] that inequality (4.3.30) is also a sufficient condition.

4.3.3 The case of a general domain In this section we assume that  is a bounded C 2 domain of RN and that 0 2 @. We recall that .x/ D dist .x, @/ and P :D P is the Poisson kernel in . The following estimate is classical (see e.g. [53]) P .x, y/  .x/jx  yjN

8.x, y/ 2  @.

(4.3.31)

Theorem 4.3.11. Assume N  2, ˛ > 2 and g 2 GQ0 satisfies lim inf r qc,˛ jxj˛ g.x, r / > 0 r !1

lim sup jr jqc,˛ jxj˛ g.x, r / < 0, r !1

(4.3.32)

essentially uniformly in a neighborhood of 0. If u 2 C 2  \ C. n ¹0º/ satisfies u C g ı u D 0

(4.3.33)

in  and is uniformly continuous on @ n ¹0º, it can be extended as a locally bounded function in .

139

Section 4.3 Boundary singularities

Proof. Without restriction we can assume that  is bounded. Since the restriction of u to @ n ¹0º is uniformly continuous, it can be extended as a continuous function

2 C.@/. We put M˙ D max@ ˙. There exist a > 0 and m > MC such that g.x, r /  ajxj˛ r qc,˛ for r  m and a.e. x 2 . By Kato’s inequality the function .u  m/C satisfies q

.u  m/C C ajxj˛ .u  m/Cc,˛  0

(4.3.34)

in  and it vanishes in a neighborhood of @ n ¹0º. For ı > 0 small enough, let wı be the solution of the problem q

wı C ajxj˛ wı c,˛ D 0

in  n Bı

wı D 0

on @ n Bı

wı D 1

on @Bı \ .

(4.3.35)

The construction is performed by a standard increasing scheme. Then wı  u  m. Since the mapping ı 7! wı is increasing, w :D limı !0 wı is a solution of w C ajxj˛ w qc,˛ D 0 wD0

in  on @ n ¹0º,

(4.3.36) q

,˛

and it dominates u  m in . For r > 0, the rescaled function wr :D Tr c,˛ Œw satisfies the same problem (4.3.36) as w, except that  is replaced by r :D r 1 . Because w satisfies (4.3.5), wr verifies the same inequality with .x/ replaced by r .x/ D dist .x, @r /. Furthermore, for any R > 0, there exists r0 > 0 such that, for any 0 < r < r0, any 0 < ı < R and any 1 < p < 1, the function wr is bounded in W 2,p .r \ .BR n B ı //. Therefore, there exist a subsequence wrk and c Q rk of wrk by zero in r converges w0 2 C.RN C n ¹0º/ such that the extension w N to w0 locally uniformly in RN C n ¹0º. Moreover w0 D 0 in @RC n ¹0º, w0 satisfies q c,˛ 2 N D 0 in RN w0 C ajxj˛ w0 C and finally w0 2 C .RC /. For any ı > 0, w0 is N dominated in RC n Bı by the minimal solution Wı of

W C ajxj˛ W qc,˛ D 0

q

,˛

in RN C n Bı

W D0

on @RN C n Bı

W D1

on RN C \ @Bı .

(4.3.37)

Furthermore, Tr c,˛ ŒWı  D Wr 1ı . If we let ı ! 0, we obtain that w  W0 :D qc,˛ ,˛ ŒW0  D W0 , it is self-similar and vanishes limı !0 Wı in RN C . Since W0 satisfies Tr N on @RC n ¹0º. By (4.3.9) and Proposition 4.3.3, W0 D 0 and similarly w0 D 0. This implies lim jxjN 1 w.x/ D 0. x!0

140

Chapter 4 Isolated singularities

Using estimates (4.3.31) and the boundary Harnack principle (see Corollary 1.6.3), we derive w.x/ lim D 0. x!0 P .x, 0/ Hence w D 0 by the maximum principle, since P .., 0/ is a supersolution for any > 0. Going back to u we conclude that u  MC in . In the same way u is bounded.  The condition q  qc,˛ is sharp since we have seen in Chapter 3 that for any y 2 @ and k  0 there exists a unique function u D uk,y 2 C 2 ./ \ C. n ¹yº/, which is a solution of (4.2.1) in , vanishes on @ n ¹yº and satisfies u.x/ D k. 3x!0 P .x, y/ lim

(4.3.38)

Furthermore the mapping k 7! uk,y is increasing and limk!1 uk,y D u1,y . The function u1,y is the unique solution of (4.2.1) in , which vanishes on @ n ¹0º and satisfies Z lim

ı !0 †ı \B˛ .y/

u.x/dS.x/ D 1

8˛ > 0.

(4.3.39)

Furthermore lim

3x!y .x  y/=jx  yj ! 

jx  yj.2C˛/=.q1/ u1,y .x/ D 1, !Q q,˛ ..x  y/=jx  yj/

(4.3.40)

Finally, for any positive solution u of (4.2.1) which is continuous in  n ¹yº and is uniformly continuous on @ n ¹yº the following dichotomy occurs: N 1 (i) either, for any  2 SC ,

lim

3x!y .x  y/=jx  yj ! 

jx  yj.2C˛/=.q1/ u.x/ D 1, !Q q,˛ ..x  y/=jx  yj/

(4.3.41)

(ii) or there exists k  0 such that lim

3x!y

If k D 0, u is the zero function.

u.x/ D k. P .x, y/

(4.3.42)

141

Section 4.3 Boundary singularities

The previous classification results extends to signed solutions and, up to a change of variable, we can assume that the singular boundary point is 0. Since  is a bounded C 2 domain of RN with 0 2 @, we can assume that there exist a neighborhood G of 0 and a C 2 real value function defined on G \ @RN C such that 0 G \ @ D ¹x D .x1, x 0 / : x 0 2 G \ @RN C , x1 D .x /º, N 1 (notice that @RN ). Furthermore .0/ D r .0/ D 0. If we set ˆ.x/ D y, C R 0 with y1 D x1  .x / and yi D xi if i D 2, ..., N , then ˆ is a C 2 diffeomorphism 2 from G to GQ D ˆ.G/, and ˆ. \ G/ D GQ \ RN C . Let  2 C ./ be a harmonic function in . If u 2 C 2 ./ \ C. n ¹0º/ coincides with  on @ n ¹0º we set

u.x/  z.x/ D u.y/, Q

z.x/ D zQ .y/,

Q .x/ D .y/,

(4.3.43)

for x D ˆ1.y/ with y 2 GQ \ RN C . If .r , / :D .jyj, y=jyj/ are the spherical coordinates, we have the following general classification result. Theorem 4.3.12. If ˛ > 2, 1 < q < qc,˛ and u 2 C. n ¹0º/ \ C 2 ./ is a solution of (4.2.1) which coincides with  2 C 2./ on @ n ¹0º. Then r .2C˛/=.q1/u.r Q , ./ q,˛ 1 N 1 converges to a connected and compact subset of EQ N 1 in the C -topology of SC when r ! 0.

SC

Proof. We divide the proof in several steps. Step 1: Straightening the boundary. We denote by r 0 the covariant derivative on S N 1 identified with the tangential gradient via the imbedding S N 1  RN and put n D y=jyj. Then 1 r uQ D uQ r n C r 0 uQ r 1 uQ y1 D uQ r n  e1 C r 0 uQ  e1 r 1 0 r uQ y1 D uQ r r n  e1n C r 0 .uQ r n  e1 / r  1   1 1  2 r 0 uQ  e1  r 0 uQ r  e1 n C 2 r 0 r 0 uQ  e1 r r r 1 0 r D r n C r

r N 1 1  D r r C

r C 2  . r r

(4.3.44)

142

Chapter 4 Isolated singularities

Plugging these expressions into (4.2.1) yields, after some lengthy computations, to   r 2uQ r r 1  2 r n  e1 C jr j2 .n  e1/2   C r uQ r N 1r ne1 2r 0 .ne1/r 0 Cr jr j2 r 0 .ne1 /e1   C b 2 r  jr j2 n  e1  r r 0 uQ  e1 (4.3.45)   C r 2n  e1 jr j2 C 2 r r 0 uQ r  e1  2r 0 uQ r  r 0 n  e1 2 C jr j2 r 0 .r 0 uQ  e1/  e1  r 0 .r 0 uQ  e1/  r 0

r Q D 0. C  uQ C r 2C˛ juQ C zQ jq1 .uQ C z/ Using the transformation t D ln r ,

t  0,

, u.r Q , / D r .2C˛/=.q1/ v.t , /, zQ .r , / D r .2C˛/=.q1/˛.t , /,

we obtain finally that v satisfies     qC1C˛ .1 C 1 / v t t C N  2 C 2 v t C N ,q,˛ C 3 v C  v q1    4 C r 0 v t  ! 5 C r 0 .r 0 v  e1 /  ! 6  jv C ˛jq1.v C ˛/ D 0, (4.3.46) C r0v  ! N 1 :D QR , where the j are expressed by on .1, ln R SC

1 D 2 r n  e1 C jr j2 .n  e1/2 2 D r n  e1 C r jr j2 r 0 .n  e1 /  e1  2r 0  r 0 .n  e1 / qC3 qC3

r n  e1  jr j2 .n  e1 /2 q1 q1 qC1 qC1 2r

r n  e1 C 2 jr j2 n  e1 C  n  e1 3 D 4 q1 q1 q1 2r 4 jr j2 r 0 .n  e1/  e1 C r 0  r 0 .n  e1 / C q  1 q  1   !  D r C 2 qC1  qC3 jr j2 n  e e C2

4

q1 r

q1

1

(4.3.47)

1

  2 !  5 D 2 r C 2jr j2 n  e1 e1  n  e1 r 0

r 2 0 !  2 6 D jr j e1 C r . r The j are uniformly continuous functions of t :D ln jxj and  2 S N 1 for j D 1, ..., 6 and they satisfy ˇ ˇ j˛.t , /j, ˇ j .t , /ˇ  C e t j D 1, ..., 6. (4.3.48)

143

Section 4.3 Boundary singularities

  Furthermore ˛.t , ./, 1 , ! 5 and ! 6 are C 1 and they satisfy ˇ ˇ ˇ ˇ j˛ t .t , /j C jr 0 ˛.t , /j, ˇ j t .t , /ˇ C ˇr 0 j .t , /ˇ  C e t

j D 1, 5, 6.

(4.3.49)

Step 2: A priori estimates. We define the non-autonomous second order differential operator L acting on C 1 .QR / by     qC1C˛ Lt Œ  D .1 C 1 / v t t C N  2 C 2 v t C N ,q,˛ C 3 v C  v q1    C r0v  ! 4 C r 0 v t  ! 5 C r 0 .rv  e1 /  ! 6 . (4.3.50) Because of (4.3.48), the operator remains uniformly elliptic for t  T0 , with continuous coefficients. By the W 2,p -regularity theory applied to problem Lt Œv C jv C ˛jq1 .v C ˛/ D 0 in QR N 1 v D 0 on @` QR :D @SC .1, ln R,

(4.3.51)

there holds, for any p 2 .1, 1/ and any T < T0  2 < ln R  2, kvkW 2,p ..T 1,T C1/S N 1/ C

   CT kvkLp ..T 2,T C2/S N 1/ C kjv C ˛jkLp ..T 2,T C2/S N 1/ . (4.3.52) C

C

Notice that CT remains uniformly bounded if T  T0, because of uniform continuity of the j . Since v remains bounded, it follows that the right-hand side of (4.3.52) remains bounded independently of T . Using the Sobolev imbedding theorems, for any 2 .0, 1/ there holds, kv.t , ./k

N 1 C 1, .SC /

C kv t .t , ./k 

C sup jhj jhj1

N 1 C 0, .SC /

kv t .t C h, ./  v t .t , ./kC.S N 1 /  C .

(4.3.53)

C

N 1 Therefore the negative trajectory TŒv of v is relatively compact in the C 2 .SC /topology and it admits a non-empty connected and compact limit set at 1 that we denote by E. N 1 , we Step 3: Energy estimates. Multiplying (4.3.46) by v t and integrating on SC obtain Z h1  i d 1 .1 C 1 /v 2t  jr 0 vj2 C N ,q,˛ v 2  jv C ˛jqC1 dS./ N 1 dt SC 2 qC1 Z Z   qC1C˛ C 2 v 2t dS.gs/ C N 2 AdS./, D N 1 N 1 q1 SC SC (4.3.54)

144

Chapter 4 Isolated singularities

where AD

1   1 t v 2t  3 vv t  v t r 0 v  ! 4  v t r 0 v t  ! 5 2

  v t r 0 .r 0 v  e1//  ! 6 C .v C ˛/jv C ˛jq1 ˛ t D

6 X

Aj .

j D1

Using (4.3.48) and (4.3.53), we obtain Z jA1 C A2 C A3 C A6 jdS./  C e t , Z

N 1 SC

1 A4 dS./ D N 1 2 SC

Z

r 0 v 2t

N 1 SC

1  ! 5 dS.gs/ D  2

Z N 1 SC

 5 dS./ v 2t div0!

ˇZ ˇ ˇ ˇ ˇ ˇ A4 dS./ˇ  C e t , ˇ ˇ S N 1 ˇ

thus

C

and Z N 1 SC

Z jA5 jdS./  C

N 1 SC

jv t jjD 2 vje t dS./ 

 C et C et   2 2 2 kv t kL D v  2 N 1 . 2 .S N 1 / C C L .SC / 2 2

Therefore, by (4.3.53), Z

T1

Z

N 1 1 SC

jA5 jdS./ D

1 Z X

T1 n

nD0 T1 n1

Z N 1 SC

jA5 jdS./  C .

Finally, since 1 < q < qc,˛ ,  lim

t !1

N 2

 qC1C˛ qC1C˛ C 2 .t , / D N  2 < 0, q1 q1

N 1 and this limit is uniform on SC . Therefore, integrating (4.3.54) and using (4.3.53), yields Z T0 Z v 2t dS./ < 1, (4.3.55) 1

N 1 SC

which, combined with the Hölder estimate in (4.3.53) implies Z v 2t dS./ D 0. lim t !1 S N 1 C

(4.3.56)

145

Section 4.3 Boundary singularities

Using again (4.3.53) and the uniform continuity of v t , we obtain that v t .t , ./ converges N 1 . to 0 when t ! 1, uniformly on SC N 1 / and  2 C01 .R/, with support in .1, 1/, Step 4: Convergence. Let  2 C01 .SC R t   0 and R ds D 1. Put  .s/ D .t  s/, then

Z

t C1 Z t 1

N 1 SC

.1 C 1 /v t t  t dS./ds Z D

t C1 Z N 1 SC

t 1

. tt .1 C 1 / C  t 1 t /v t dS./ds ! 0

by (4.3.49), when t ! 1. Moreover Z t C1 Z   qC1C˛ C 2 v t  t dS./ds ! 0, N 2 N 1 q1 t 1 SC Z

t C1 Z N 1 SC

t 1

and Z

t C1 Z t 1

N 1 SC



   r0v  ! 4 C r 0 v t  ! 5  t dS./ds ! 0

 r 0 .r 0 v  e1/  ! 6  t dS./ds Z D

t C1 Z t 1

N 1 SC

    6 C ! r 0 v  e1  div0! 6 div0   t dS./ds ! 0,

by (4.3.48), (4.3.49). Therefore, if ! belongs to the limit set E of the negative trajectory, there exists a sequence ¹tnº converging to 1 such that v.tn, ./ ! ! in the N 1 /-topology. Then kv.t , ./  !kC 0 .S N 1/ ! 0 uniformly on Œtn  1, tn C 1 C 1 .SC C when n ! 1. Thus Z tn C1 Z   N ,q,˛ v C  v  tn dS./ds tn 1

N 1 SC

Z D

tn C1 Z

 N 1 SC

tn 1

Z

! Furthermore Z tn C1 Z tn 1

q1

N 1 SC

 N ,q,˛ v C v   tn dS./ds

jv C ˛j

N 1 SC

  N ,q,˛ ! C !  dS./ as n ! 1.

tn

.v C ˛/ dS./ds !

Z N 1 SC

j!jq1 !dS./

146

Chapter 4 Isolated singularities

N 1 Consequently ! 2 C 1 .SC / satisfies R   q1 ! dS./ D 0 S N 1 N ,q,˛ ! C !   j!j C

q,˛

Therefore ! belongs to ES N 1 . C

N 1 8 2 C01 .SC /. (4.3.57) 

Using the structure of the set EQ S N 1 detailed in Proposition 4.3.3 we obtain in parq,˛ C

ticular:

Corollary 4.3.13. Under the assumptions of Theorem 4.3.12, the limit set is reduced to a single element if one of the following conditions holds: (i) 1 < q < qc,˛ and u has constant sign.  q < qc,˛ . (ii) qi ,˛ :D N C2C˛ N (iii) N D 2. By combining the ideas used in the proof of Theorem 4.2.10 (in particular the function) with the energy method as in the proof of Theorem 4.3.12, we can describe, in many cases, the behavior of a solution u when limr !0 r .2C˛/=.q1/u.r , ./ D 0 (see [54] for a detailed analysis). Theorem 4.3.14. Let the assumptions of Theorem 4.3.12 be fulfilled and assume that u satisfies limr !0 r .2C˛/=.q1/ u.r , ./ D 0. Then there exist an integer k in the range 1  k < .2q C ˛/=.q  1/  N and a non-zero k 2 ker. C k I / where k D N 1 /, such that k.N C k  2/ is the k-th eigenvalue of  in W01,2.SC lim r N Ck2u.r Q , ./ D k ,

(4.3.58)

r !0

if one of the following conditions is satisfied: (i)

u has constant sign.

(ii) N=2. (iii) qi ,˛  q < qc,˛ . (iv) N  3 and N ,q,˛ … . bS N 1 / D ¹`.N C `  2/ : ` 2 N º. C

q,˛ Remark 4.9. It would be interesting to prove that for any signed ! 2 EQ N 1 , there SC

exists a solution of u of (4.2.1) continuous in  n ¹0º, which vanishes on @ n ¹0º and satisfies 2C˛ (4.3.59) limr !0 r q1 u.r Q , ./ D !, where uQ is defined in (4.3.43). When  D RN C \BR some solutions can be constructed by using the symmetry of the domain with respect to the plane x2 D 0. These solutions

147

Section 4.4 Boundary singularities with fading absorption

vanish on the symmetry plane. The same question can be raised for solutions with weak singularities satisfying (4.3.58), and the same partial result of existence can be given when  D RN C \ BR .

4.4 Boundary singularities with fading absorption 4.4.1 Power-type degeneracy In this section we present results concerning boundary singularities of signed solutions of u C  ˛ jujq1 u D 0 (4.4.1) where ˛ > 2 and q > 1, which extend what some of the results of Section 3.4. As a model case we consider the equation u C x1˛ jujq1 u D 0, 0

(4.4.2)

:D ¹x D .x1 , x / 2 R : x1 > 0º. We recall that x 0 D in the half-space .x2 , ..., xN /. We express the spherical coordinates in RN C in order x1 D jxj cos D jxj.  e1/ with 0   =2. If we look for separable solutions of (4.4.2) of the form (4.3.9) vanishing on @RN C n ¹0º, then ! satisfies RN C

N

 !  N ,q,˛ ! C . . e1 /˛ j!jq1 ! D 0 !D0

N 1 in SC N 1 on @SC .

(4.4.3)

N 1 N 1 Let E q,˛ /\C 2 .SC / satisfying (4.4.3). Then E q,˛ N 1 be the set of ! 2 C.SC N 1 has SC

SC

q,˛ a structure similar to the one of EQ N 1 described by Proposition 4.3.3. The proofs are SC

essentially the same except for existence in which case it is presented in Lemma 3.4.7. Proposition 4.4.1. (i)

If q  qc,˛ :D .N C 1 C ˛/=.N  1/, E

q,˛ N 1 SC

D ¹0º.

q,˛

(ii) If 1 < q < qc,˛ , ES N 1 admits a unique positive element !q,˛ . As a consequence q,˛ E N 1 SC

C

\

N 1 CC.SC /

D ¹0, !q,˛ º.

(iii) If qi ,˛ :D .N C 2 C ˛/=N  q < qc,˛ , E q,˛ N 1 D ¹!q,˛ , !q,˛ , 0º. SC

(iv) If 1 < q < qi ,˛ , E

q,˛ N 1 SC

contains signed elements besides 0 and ˙!q,˛ .

If ˛ > 2, q > 1 and u 2 C 2./ \ C. n ¹0º/ is a solution of (4.4.1) which vanishes on @ n ¹0º then the estimate obtained in Proposition 3.4.4 reads as follows ju.x/j  C jxj.qC1C˛/=.q1/ .x/

8x 2  n ¹0º.

The following regularity result is needed since ˛ may be negative.

(4.4.4)

148

Chapter 4 Isolated singularities

Lemma 4.4.2. Let R > 0, a 2 @ such that jaj > R, GR .a/ D  \ BR .a/, q > 1, ˛ > 2 and  2 R. Assume that v 2 C.GR .a// \ W 1,2 .GR .a// satisfies v  v C  ˛ jvjq1 v D 0 vD0

in GR .a/ on @GR .a/.

(4.4.5)

Then v 2 C 1, .GR0 .a// \ W 2,s .GR0 .a// for any 0 < R0 < R, some 2 .0, 1/ and s > 1. Proof. As in the proof of Proposition 4.4.1 (i), there exists C > 0 such that C kv kL1 .S N 1/  v  C kvC kL1 .S N 1/ .

(4.4.6)

Thus  ˛ jvjq  C ˛Cq . Clearly, if ˛ C q  0 the standard theory applies and yields v 2 W 2,p .GR0 .a// for any R0 < R, 1 < p < 1 and thus v 2 C 1, .GR0 .a// for any < 1. Next we suppose 1 < ˛ C q < 0. Since  ˛Cq 2 Ls .GR .a// for any 1 s 2 .1, ˛Cq /, it follows that v 2 W 2,s .GR0 .a//. For proving the C 1, regularity, the problem can be reduced to the case where the boundary is flat, that is if vQ satisfies in B1C

vQ D x1˛Cq

on @B1C ,

vQ D 0

(4.4.7)

where is bounded and is C 1. Set .˛CqC1/

.x1, x 0 / D x1

Z

x1

s ˛Cq .x 0 , s/ds,

0

and denote by w the solution of w D x1˛CqC1

in B1C on @RN C \ B1 ,

wx 1 D 0

(4.4.8)

on RN C \ @B1

wDh

where h is a smooth function defined in B2 . By Schauder’s estimates D 2w is C  with

D ˛ C q C 1 2 .0, 1/. Differentiating with respect to x1 we obtain ˛Cq

wx1 D x1



in B1C

wx 1 D 0

on @RN C \ B1 ,

wx1 D hx1

on RN C \ @B1 .

Since vQ  wx1 is smooth, the result follows.

(4.4.9)



149

Section 4.4 Boundary singularities with fading absorption

Remark 4.10. The same argument applies to problem (4.4.3) and any element of q,˛ N 1 N 1 N 1 /\C 1, .SC / if 1 < ˛Cq < 0, and to W 2,p .SC / E N 1 belongs to W 2,s .SC SC

if ˛ C q  0. The description of isolated singularities for (4.4.1) is very similar to the one of (4.2.1) and we use the same notation uQ for the function obtained by (4.3.43) when straightening the boundary (see Section 4.3.3). Theorem 4.4.3. Assume that  is a C 2 domain, 0 2 @ is in normal position, q > 1 and ˛ > 2. If u 2 C 2 ./ \ C. n ¹0º/ is a solution of (4.4.1) which coincides with  2 C 2 ./, then r .2C˛/=.q1/u.r Q , ./ converges to a connected compact subset of q,˛ 1 N 1 E N 1 in the C .SC /-topology when r ! 0. SC

Proof. We straighten the boundary and obtain that .x/ D c.x/.x1  .x 0 // with c 2 C 1 . \ BR / satisfies 0  c.x/  1 and lim c.x/ D 1 , lim rc.x/ D 0.

x!0

x!0

(4.4.10)

Q , / satisfies We set t D ln r , then v.t , / D r .2C˛/=.q1/u.r      v t C N ,q,˛ C 3 v C  v C r 0 v  ! C 4 .1 C 1 / v t t C N  2 qC1C˛ 2 q1   C r 0vt  ! 5 C r 0 .r 0 v  e1/  ! 6  c1 .  e1/˛ jv C ˛jq1.v C ˛/ D 0, (4.4.11) N 1 on .1, ln R SC :D QR , where the j and ˛ are already defined in Theorem 4.3.12 and c1 is a C 1 function which satisfies c1 .y/ D 1 C O.e 2t /

jDc1 .y/j D O.e t /

when t ! 1.

(4.4.12)

Mutatis mutandis the proof is very similar to the one of Theorem 4.3.12 although some modifications are needed since it may not be true that D 2 v remains bounded in N 1 / independently of T , but only in W 2,s ..T  1, T C 1/ W 2,2 ..T  1, T C 1/ SC N 1 SC / for some s > 1. The adaptation of the estimate of the term A5 in the proof of Step 3 in Theorem 4.3.12 is made as follows: Z Z jA5 jdS  C jv t jjD 2 vje t dS N 1 SC

N 1 SC



 C et  C et  2 s s kv t kL D v  s N 1 . s .S N 1 / C L .S / s s

Since v t remains bounded in QR , the integrability of A5 in QR follows.



150

Chapter 4 Isolated singularities

Using the structure of E

q,˛ N 1 SC

described in Proposition 4.4.1, we obtain:

Corollary 4.4.4. Let u be a non-negative solution. (I) If 1 < q < qc,˛ , there holds: 2C˛

Q , ./ D 0, (i) either limr !0 r q1 u.r 2C˛

(ii) or limr !0 r q1 u.r Q , ./ D !q,˛ ../. (II) If qi ,˛ :D

N C2C˛ N

(i) either limr !0 r (ii) or limr !0 r (iii) or limr !0 r

 q < qc,˛ , there holds:

2C˛ q1

u.r Q , ./ D 0,

2C˛ q1

u.r Q , ./ D !q,˛ ../,

2C˛ q1

u.r Q , ./ D !q,˛ ../.

N 1 /-topology. Furthermore, all the above convergences hold in the C 1.SC 2C˛

When limx!0 jxj q1 u.x/ D 0, in most of the cases the solution exhibits a weak singularity at 0, and Theorem 4.3.6 remains valid except that the reference is to Theorem 4.4.3. Notice also that if  D RN C and .x/ D x1 , all the statements of Theorem 4.3.7, Theorem 4.3.8, Lemma 4.3.9 and Proposition 4.3.10 are still valid provided we replace equation (4.2.1) by (4.4.2).

4.4.2 A strongly fading absorption Let  be a C 2 bounded domain whose boundary contains 0. If one replace  ˛ in (4.4.1) by a general `./ where ` is positive in .0, 1/, vanishes at 0 and is very flat near 0, it may happen that there exists no barrier (see Definition 3.1.10) at any point on @ and strong isolated singularities propagate along the boundary. This is the case if `.x/ D e =.x/ with > 0. Since e = .P .., 0//q 2 L1 ./, for any k  0, there exists a unique solution u D uk,0 of

u C e   jujq1 u D 0 u D kı0

in 

(4.4.13)

on @,

by Theorem 2.3.7. By the maximum principle k 7! uk,0 is increasing and locally bounded in  by the local Keller–Osserman condition. Then there exists u1,0 D limk!1 uk,0 . The next result shows the propagation of the singularity from 0 to the whole boundary. Theorem 4.4.5. The function u1,0 is a positive solution of

u C e   jujq1 u D 0

in 

(4.4.14)

151

Section 4.4 Boundary singularities with fading absorption

which satisfies lim u1,0.x/ D 1.

(4.4.15)

.x/!0

Furthermore it is the minimal solution satisfying (4.4.15). Proof. We can assume that the hyperplane H0 D ¹x D .0, x 0 /º is tangent to @ at 0. We write exp.  / D h ı . Step 1: The case 1 < q < .N C 1/=.N  1/. For 0 <  ˇ0 and m > 0 small 1 enough, we set `. / D h q1 . /. By the maximum principle u1,0 is bounded from below by `. /U in m  D ¹x 2  : jxj < m , 0 < .x/ < º, where U is the unique solution of v C v q D 0 in m  , v D 1ı0 on @m  . There holds lim jxj2=.q1/ U .x/ D !.x= jxj/,

(4.4.16)

x!0 x 2 m 

where ! is the unique positive solution of  !  N ,q,˛ ! C ! q D 0 !D0

N 1 in SC ,

on

(4.4.17)

N 1 @SC .

If we write U .x/ D 2=.q1/U1, .x= / D 2=.q1/ U1, .y/, then U1, satisfies

q

U1, D U1,

y D x= ,

in Dm :D 1m  .

When ! 0, Dm converges to D0 D .0, 1/ RN 1 . Thus, for any 0 < 1 < 2 < 1, there exists 0 such that if 0 <  0 , the following inclusion holds ˇ ˇ Gm :D ¹y D .y1 , y 0 / : 1 < y1 < 2 , 1 < ˇy 0 ˇ  m=2 º  Dm . Because

U1, .y/  C jyj2=.q1/ ,

U1, converges uniformly to some U1 on any compact subset of D0 n ¹0º as ! 0, and U1 is the unique function which verifies q

U1 D U1

in D0 ,

vanishes on @D0 n ¹0º and satisfies lim jyj2=.q1/ U1 .y/ D !.y= jyj/.

y!0 y 2 D0

(4.4.18)

152

Chapter 4 Isolated singularities

Therefore there exists some  2 .0, 1/ such that  .y   /  ˇ ˇ 1 1 , for y1 2 Œ1 , 2  and ˇy 0 ˇ D 1. (4.4.19) U1, .y1 , y 0 /   sin 2  1 Notice that the function y1 7! y1 D 1 and for y1 D 2.

 .y1 /

D sin..y1  1/=.2  1 // vanishes for

We first suppose that N D 2. Then for any ˇ > 0 the function   m  jy 0 j sinh ˇ 2 0 0  m  y 7! ˇ .y / D sinh ˇ 2 1

(4.4.20)

is non-negative, takes the value 1 for jy 0 j D 1 and vanishes for jy 0 j D m=.2 /. If we set ,ˇ .y1 , y 0 / D   .y1 / ˇ .y 0 /, there holds  ,ˇ D ˇ 2  Since ,ˇ  , it follows that  ,ˇ  ˇ 2 

 2 ,ˇ . .2  1/2

 2 q 1q ,ˇ .2  1 /2

(4.4.21)

in Gm ,

(4.4.22)

and

for y1 D i , i D 1, 2, ,ˇ .y1 , y 0 / D 0 ,ˇ .y1 , y 0 / D 0 for jy 0 j D m=2 0 ,ˇ .y1 , y / D   .y1 / for jy 0 j D 1.   2 1q D 1. By (4.4.19) and the maximum We choose ˇ such that ˇ 2  .  2  / 2 1 principle one obtains U1, .y1 , y 0 /  ,ˇ .y1 , y 0 / in Gm .

(4.4.23)

Therefore u1,0 .x1, x 0 /  `. / 2=.q1/ U1, .x1 = , x 0 = /  `. / 2=.q1/ ,ˇ .x1 = , x 0= /, for 1  x1  1 and  jx 0 j  m=2. Take x1 D .1 C 2/=2 D  , then    0 j sinh ˇ m2jx 2  m  . u1,0 . , x 0 /  `. / 2=.q1/ sinh ˇ 2  1 If jxj0  m=4,

   0j sinh ˇ m2jx 0 2  m  D e ˇ.1jxj =/ .1 C ı.1// as ! 0. sinh ˇ 2  1

153

Section 4.4 Boundary singularities with fading absorption

Thus

0

u1,0. , x 0 /  `. / 2=.q1/e ˇ e ˇ jx j= .1 C ı.1//,

and

0

lim inf u1,0 . , x 0 /  e ˇ lim inf `. / 2=.q1/ e ˇ jx j= . !0

Since `. / D

!0

e =..q1// ,

i h  0 `. / 2=.q1/ e ˇ jxj =/  2=.q1/ exp 1 =.q  1/  ˇjxj0 . If we fix jxj0 < ˇ =.q  1/, then lim inf u1,0. , x 0 / D lim u1,0.x/ D 1, !0

.x/!0

and this limit is uniform on any compact subset of ¹x 0 : jxj0 < ˇ =.q  1/º, which is equivalent to any compact of ¹x : j.x/j < ˇ =.q  1/º. Put  D ˇ =.2.q  1//. Because this blow up holds in a fixed neighborhood @ \ BN  .0/ of 0, we can replace 0 by any point P in @ \ BN  .0/ and conclude that lim u1,0.x/ D 1,

.x/!0

(4.4.24)

uniformly if j.x/  P j  . By iterating this process we infer that lim u1,0.x/ D 1.

.x/!0

(4.4.25)

Next we assume N  3. Let ˇ > 0 to be fixed and ˇ ˇ  D ¹y 0 2 RN 1 : 1 < ˇy 0 ˇ < m=2 º and let bˇ .y 0 / be the solution of y 0 bˇ D ˇ 2bˇ bˇ .y 0 / D 1 bˇ .y 0 / D 0

in if if

 , ˇ 0 ˇ ˇy ˇ D 1, ˇ 0ˇ ˇy ˇ D m=2 .

(4.4.26)

The function ,ˇ .y1 , y 0 / D   .y1 /bˇ .y 0 / satisfies also (4.4.21) in Gm . Therefore, if we choose ˇ as in the case N D 2, (4.4.23) is still valid. Since bˇ is a Bessel function, its behavior at infinity is well known (see e.g. [124, Ch. XVII]): for jxj0  m=4, there holds  0 1N=2 0 jxj bˇ .y/ D Cˇ e ˇ jxj = .1 C ı.1//, as ! 0. We conclude as in the case N D 2.

154

Chapter 4 Isolated singularities

Step 2: The case q  .N C 1/=.N  1/. Let ˛ > 0 such that q < qc,˛ :D .N C 1 C ˛/=.N  1/. We write

Q h..x// D  ˛ .x/h..x//,

with

Q h..x// D  ˛ .x/h..x//.

Q / is non-decreasing near r D 0 and we extend it by We can assume that r 7! h.r Q continuity at r D 0 by putting h.0/ D 0. Thus there holds ˛ Q .x/uq u  h. /

in  :D ¹x 2  : .x/ < º.

The equation U C  ˛ .x/U q D 0 admits weak and strong isolated singularities on the boundary; any positive solution with a strong singularity at x D 0 satisfies lim jxj.2C˛/=.q1/ U .x/ D !q,˛ .x= jxj/,

x!0 x 2 

(4.4.27)

where !q,˛ is the unique positive element of E q,˛ N 1 . Then we proceed as in Step 1, SC

with some minor changes of coefficients, to derive (4.4.24) and then (4.4.25). Step 3: End of the proof. The minimal solution u of (4.4.14) is constructed by considering the increasing sequence un of solutions of

un C e   uqn D 0 un .x/ D n

in , on @.

(4.4.28)

When k ! 1, un ! u, thus u1,0  u. On the other hand, ukı0 is constructed by approximating the Dirac mass on the boundary by bounded functions gm,k . Thus the corresponding solutions ugm,k of (4.4.14) are all dominated by un for n large enough, and therefore by u. This implies u1,0 D u.  The following result shows that the boundary trace (see Chapter 3) of solutions of (4.4.14) cannot be any type of Borel measure. Corollary 4.4.6. Let 1 < q < .N C 1/=.N  1/ and > 0. If u is a positive solution of (4.4.14), then it admits a boundary trace which is either the measure 1 which satisfies 1 .E/ D 1 for any non-empty Borel set E  @, or any finite positive Borel measure.

155

Section 4.4 Boundary singularities with fading absorption

Proof. If  is any positive bounded Borel measure on @, e   .P Œ/q 2 L1.; /. Therefore it is an admissible measure (see Definition 2.2.5) and any positive finite Borel measure can be a boundary trace by Corollary 2.2.6. Notice that this result is valid even if q  .N C 1/=.N  1/. By Proposition 3.1.6, the boundary trace is a bounded Borel measure if and only if   q e u 2 L1.; /. Let us assume that S.u/ ¤ ; and Z

e   uq dx D 1. (4.4.29) 

Since it is not possible to construct a barrier (see Section 3.1.2), we proceed as in the proof of Theorem 6.1.14: if (4.4.29) holds, it implies Z lim sup udS D 1. (4.4.30) !0

†

where † D ¹x 2  : .x/ D º. We construct a sequence ¹ nº converging to 0 and a sequence of points ¹anº such that an 2 †n , an ! a 2 @ and Z udS D kn ! 1 as n ! 1. (4.4.31) Bn .an /\†n

For arbitrary k and n large enough, there exists m D m.k, n/ such that Z min.m, u/dS D k,

(4.4.32)

Bn .an /\†n

and we set n D min.m, u/Bn .an /\†n . By the maximum principle, u is bounded from below in 0n D ¹x 2  : .x/ > n º by the solution vn of

vn C e   vn D 0 vn D n q

in 0n on †n .

(4.4.33)

When n ! 1, n * kıa in the weak sense of measure. Since q is subcritical, vn ! uk,a , locally uniformly. Thus u  uk,a in . Since k is arbitrary u  u1,a D u. This implies the claim.  Remark 4.11. If problem (4.4.13) is replaced by u C e





jujq1 u D 0 u D kı0

in 

(4.4.34)

on @

with  2 .0, 1/, it is proved in [105] that u1,0 is a solution which belongs to C.n¹0º/ and vanishes on @ n ¹0º. The precise result is actually more general and its proof is obtained by a very elaborate local energy method.

156

Chapter 4 Isolated singularities

4.5 Miscellaneous In this paragraph, we present without proof some results dealing with the description of isolated singularities of positive solutions of u C g ı u D 0

(4.5.1)

for various types of nonlinearities.

4.5.1 General results of isotropy Let  be a domain containing zero. If u is a harmonic function in  n ¹0º, it admits an expansion in series of spherical harmonics u.x/ D u.r , / D

1 X

r k k ./ C a0 N .r / C

kD0

1 X

r 2N k Q k ./,

(4.5.2)

kD1

where the k and Q k are eigenfunctions of the Laplace–Beltrami operator  on S N 1 , with corresponding eigenvalue k D k.N C k  1/. Therefore, a necessary and sufficient condition for u to present a singularity at 0 is that a0 or one of the Q k is not zero. This singularity is asymptotically radial, or isotropic, if a0 ¤ 0 and Qk D 0 for all k  1. The second statement is fulfilled if lim r N 1 ku.r , ./  u.r N /kL1.S N 1/ D 0

r !0

(4.5.3)

where u.r N / is the spherical average of u.r , ./ on S N 1 . This situation has been extended to monotone perturbation of the Laplacian. Theorem 4.5.1. Let g 2 C.R/ be non-decreasing and u 2 C 2 . n ¹0º/ a solution of (4.5.1) which satisfies (4.5.3). Then there exists c 2 R [ ¹˙1º such that u.x/ D c. x!0 N .x/ lim

(4.5.4)

Sketch of the proof. Using the equation in spherical coordinates, ur r C

N 1 1 ur C 2  u D g ı u in .0, a/ S N 1 , r r

(4.5.5)

the key point is to prove ku.r , ./  u.r N /kL1 .S N 1 / D O.r / as r ! 0. This is obtained by using the monotonicity of g, which implies Z N D .g ı u  g ı u/ N .u  u/d N  0, .g ı u  g ı u/ .u  u/d S N 1

(4.5.6)

(4.5.7)

157

Section 4.5 Miscellaneous

and the inequality Z S N 1

. .u  u/.u N  u/d N  .1  N /

N 1 .u  u/ N 2 d. r

Therefore X.r / D ku.r , ./  u.r N /kL2.S N 1 / satisfies the differential inequality X 00 C

N 1 0 N 1 X  X  0. r r2

(4.5.8)

Using Poisson formula, .4.5.3/ implies N /kL2 .S N 1/ D 0. lim r N 1 ku.r , ./  u.r

r !0

(4.5.9)

N /kL2 .S N 1/ D 0 and Combined with .4.5.8/ it implies limr !0 r 1 ku.r , ./  u.r thus .4.5.6/ follows. The proof of .4.5.4/ is easily obtained by a contradiction argument.  The same method applies for studying the behavior when x ! 1 of solutions of (4.5.1) defined in the complement of a compact set in RN .

4.5.2 Isolated singularities of supersolutions Another type of property of isotropy deals with non-negative solutions of differential inequalities such as u C g ı u  0. (4.5.10) The following result is proved in [103]. Theorem 4.5.2. Assume that N  3 and g 2 C.R/ with g.0/  0 is non-decreasing and satisfies Z 1

g.r 2N /r 1N dr < 1.

(4.5.11)

0

If u 2 C 2 . n ¹0º/ is a non-negative solution of (4.5.10). Then the following dichotomy holds: (i) either r N 2u.r , ./ converges in measure on S N 1 to some c 2 Œ0, 1/. (ii) or limx!0 jxjN 2 u.x/ D 1. Condition (4.5.11) already used in Theorem 2.3.7 with ˇ D 0, is a necessary and sufficient condition in order that for any 2 .0, 1/ there exists a solution u D u of (4.5.1) satisfying lim jxjN 2 u .x/ D (4.5.12) x!0

158

Chapter 4 Isolated singularities

(see e.g. [117]). Then the result is obtained first by introducing v D min.u, u /, which happens to be a supersolution of (4.5.1). Since g.v / is locally integrable, there exists c such that lim jxjN 2 v .x/ D c , (4.5.13) x!0

and 0  c  . By studying the monotonicity properties of 7! c and its limit when

! 1 we obtain the dichotomy in the statement of the theorem. As an application of Theorem 4.5.2, the next result can be easily derived. Corollary 4.5.3. Assume that N and g are as in Theorem 4.5.2 and u 2 C 2 . n ¹0º/ is a non-negative solution of (4.5.1). Then there exists c 2 Œ0, 1 such that (4.5.4) holds. In the two-dimensional case, condition (4.5.11) is meaningless and comparison has to be made with exponential functions. The exponential order of growth of g at 1 is defined by Z 1 ° ± g.s/e as ds < 1 . (4.5.14) agC :D inf a  0 : 0

For any 2 Œ0, 2=agC there exists a solution of (4.5.1) in  n ¹0º \ BR (R > 0) satisfying u .x/ D , (4.5.15) lim x!0 2 .x/ while no such solution can exist, even locally, if > 2=agC (see e.g. [112]). By the same sweeping method as the one used for the proof of Theorem 4.5.2, the following statement can be proved. Theorem 4.5.4. Assume that N D 2 and g 2 C.R/ is a non-decreasing function, positive if r  a. Let u 2 C 2 . n ¹0º/ be a non-negative solution of (4.5.10). If agC D 0, the conclusion of Theorem 4.5.2 is valid. If agC > 0 the following dichotomy holds: (i) either u.r , ./= ln.1=r / converges in measure on S 1 to some c 2 Œ0, 2=afC /, (ii) or lim infx!0 u.x/= ln.1=jxj/  2=afC. As a first application of Corollary 4.5.3 we consider u C e ˛u D 0

(4.5.16)

where ˛ > 0. By Corollary 4.2.2 isolated singularities of solutions are removable in dimension N  3, but in dimension 2 isolated singularities can be described. Theorem 4.5.5. Assume that N D 2 and u 2 C 2 . n ¹0º/ is a non-negative solution of (4.5.16) in  n ¹0º. Then there exists c 2 Œ0, 2=˛ such that (4.5.4) occurs.

159

Section 4.6 Notes and comments

An extension of this method to equations with forcing nonlinearity such as u D e ˛u

(4.5.17)

where ˛ > 0 yields the following result in dimension 2: Theorem 4.5.6. Assume that N D 2 and u 2 C 2. n ¹0º/ is a non-negative solution of (4.5.17) in  n ¹0º. Then there exists c 2 Œ0, 2=˛/ such that (4.5.4) occurs. In higher dimensions the problem becomes much more difficult.

4.6 Notes and comments The universal estimates of Section 4.1 were proved separately by J. B. Keller [60] and R. Osserman [96] and the construction of a maximal solution followed from this in a natural way. Later on, these estimates were rediscovered and popularized by H. Brezis and E. H. Lieb [24] in their study of the Thomas–Fermi theory which plays an important role in the modeling of the electric field potential created by the nuclear charges and distribution of electrons in an atom. The radial equation p 2 u00  u0 C u3 D 0 (4.6.1) r belongs to the class of Emden–Fowler equations (4.1.34). Precise asymptotic expansions of solutions near the origin have been obtained by A. Sommerfeld [108], and analytic properties of the solutions were proved by E. Hille [57]. Emden–Fowler equations (4.1.34), also known as the Lane–Emden equations, are the Poisson equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. In that case u is ln.1=/ where  is the density of a gas. The name came from the two astrophysicists J. H. Lane in 1870 and R. Emden in 1897 who introduced them. In their case, N D 3 and q integer, and only numerical and graphical results were obtained. The first real mathematical study, always in the radial case, is due to R. H. Fowler who in 1931 made a general analysis of the asymptotic behavior of the solutions of u00 D ˙t  un ,

(4.6.2)

a form that satisfies the radial solutions of (4.1.34) after a suitable change of variables. His method was strongly based upon some former result due to G. H. Hardy which lists all the possible expressions of the asymptotics of the solutions when n is a rational number. Because of its role in astrophysics a much more detailed numerical analysis of radial solutions of 2 (4.6.3) u00  u0 C e u D 0 r was made by S. Chandrasekhar in his classical book [28]. When N D 2, the equations associated to inequalities (4.1.26) (also (4.1.27)) are associated to a conformal change

160

Chapter 4 Isolated singularities

of metric with a fixed negative Gaussian curvature; it is sometimes called the Liouville equation and was studied by L. Bieberbach [18]. When N D 3, a uniqueness result for radial solutions was obtained by R. Rademacher [100]. For general solutions of semilinear equations with superlinear absorption, the first result of removability is contained in H. Brezis and L. Véron’s paper [27] in which Theorem 4.2.1 and Corollary 4.2.2 (with ˛ D 0) are proved. Theorem 4.2.4 is due to L. Véron [117], although a version of the result given by Theorem 4.2.5 was obtained in a more general setting by Ph. Benilan and H. Brezis in 1977. The classification of positive isolated singularities of solutions of Emden–Fowler equations (4.1.34) stated in Theorem 4.2.7 with ˛ D 0 was obtained by L. Véron [114, 115]. His method applied to positive solutions and to signed solutions provided q  .N C 1/=.N  1/. For positive solutions, his proof used the uniform Harnack inequality and comparison with radial solutions. For signed solutions, the proof relies heavily on showing that ku.r , ./  u.r N /kL1 .S N 1 / D o.r 1N / where u.r N / is the spherical average of .r , / 7! u.r , /. The general result of Theorem 4.2.9 based upon the use of the energy method and reduction to the autonomous second order equation (4.2.44) is proved by X. Y. Chen, H. Matano and L. Véron [30]. In this form, t (=ln r ) is interpreted as the “time” variable. A similar approach has been used by B. Gidas and J. Spruck [51] for analyzing the local behavior and singularities of positive solutions of u C uq D 0,

(4.6.4)

(1 < q < 1). However, in [30] the fact that the exterior problem for (4.2.44) has a unique solution allowed to introduce a semigroup in the t variable. Techniques inherited from the theory of infinite dimensional dynamical systems and due to H. Matano [88, 89] were used for showing the existence of global solutions. They are soq,0 lutions defined on R S N 1 which connect two elements of EQ S N 1 . In the same paper q,0 it is proved that if N D 2, the limit set of a trajectory is a particular element of EQ S 1 . The method of proof of this result is an adaptation of the Sturm intersection argument based upon the Jordan separation theorem. The characterization of weak singularities stated in Theorem 4.2.10 is also proved in the article. It remains an open problem to prove the convergence result in higher dimensions. One important result concerning the asymptotic behavior of solutions of v t t C av t C g v  f .v/ D 0 in RC M

(4.6.5)

where a ¤ 0, .M , g/ is a compact manifold and g the Laplacian on M , is due to L. Simon [106, 107] who proved in 1983–1985 that any bounded solution of (4.6.5) converges to a solution w of g w  f .w/ D 0 in M ,

(4.6.6)

provided f is a real analytic function. The proof is based upon an extension of the Lojaciewicz inequalities [68] concerning semi-analytic sets which state that the energy

161

Section 4.6 Notes and comments

functional 1 E.w/ D 2

Z M

  jrwj2 C 2F .w/ dvg

Z

w

where F .w/ D

f .s/ds,

(4.6.7)

0

– thus E 0 .w/ D g w  f .w/ – satisfies  0  E .w C /  jE.w C /  E. /j1 ,

(4.6.8)

for some  2 .0, 1=2/ and some suitable norm k.k. An interesting extension of this technique is proved by A. Haraux, M. A. Jendoubi and O. Kavian in [55] who obtained asymptotic stabilization of solutions of some semilinear heat equations without the analyticity assumption on f . Boundary singularities for nonlinear elliptic equations of Emden–Fowler type and the solutions of the equation (4.1.34) with ˛ D 0 have been classified by A. Gmira and L. Véron [54]. Since the classification can be reduced to the study of the longtime behavior of the solutions of the asymptotically autonomous equation (4.3.46), the same question of asymptotic stabilization in the framework of the Lojiaciewicz– Simon approach should be considered. In the same paper they also proved the existence of solutions of nonlinear boundary value problems with measures in the subcritical case. Equations with fading absorption (4.4.1) were studied by M. Marcus and L. Véron [80]. The a priori estimate (4.4.4) is derived from a result of Y. Du and Z. Guo [38]. The phenomenon of propagation of the singularity along the boundary which occurs in the case of a strongly fading absorption have been obtained by M. Marcus and L. Véron [81]. In [105], A. Shishkov and L. Véron obtained a sufficient condition for the non-propagation of the singularity along the boundary. The method which they used is based upon local energy estimates. The isotropy of singular behavior in Theorem 4.5.1 was proved by J. L. Vàzquez and L. Véron using techniques already introduced in [115]. The sweeping method for analyzing isolated singularities of positive superharmonic functions have been introduced by Y. Richard and L. Véron [103]. It is an important tool for characterizing isolated singularities of positive solutions of equations for which the Harnack inequality and the energy method are ineffective.

Chapter 5

Classical theory of maximal and large solutions

This chapter is devoted to the study of large solutions of semilinear elliptic equations in a domain of RN , which means solutions that tend to infinity at the boundary. A maximal solution is constructed under fairly general assumptions. The two main questions under review are whether the maximal solution is a large solution or not and, if so, if large solutions are unique.

5.1 Maximal solutions 5.1.1 Global conditions The construction of the maximal solutions of u C g ı u D 0

(5.1.1)

in   RN is based upon various forms of the Keller–Osserman conditions. If g is continuous in  R, any locally bounded solution is indeed a strong solution in the 2,p sense that it belongs to Wloc ./ \ C 1,˛ ./ for any 1  p < 1 and 0 < ˛ < 1. Furthermore, the equation is satisfied in the strong sense. Theorem 5.1.1. Let  be a proper domain in RN and g 2 GQ0 satisfy the extended Keller–Osserman condition. Assume also that g.x, ./ is non-decreasing. Then either .5.1.1/ admits a maximal solution in  or no solution at all exists. Proof. There exists a sequence of smooth bounded domains n satisfying n  n  nC1 and [n n D . Such a sequence is called a smooth exhaustion of  . For each n and k in N , we denote by un,k :D u the solution of u C g ı u D 0 in n u D k on @n .

(5.1.2)

Since g.x, ./ is non-decreasing, un,k  un,kC1 . Furthermore, un,k  f .n .x// follows from Theorem 4.1.2, where n.x/ D dist .x, @n /. Therefore ¹un,k º is increasing with k and converges, when k ! 1, to some locally bounded function un which is therefore a solution of .5.1.1/ in n which satisfies lim@ .x/ !0 un .x/ D 1. n By the maximum principle in n , it follows that un,k  unC1,k bn . The mapping n 7! un,k is decreasing, thus un  unC1 bn . If U 2 C 1 ./ is a solution of .5.1.1/ in  (if there is one), then un  U in n . Thus there exists u D limn!1 un . Since u is

163

Section 5.1 Maximal solutions

locally bounded, it is a solution of .5.1.1/. Furthermore, since for any n the function un dominates the restriction to n of any solution u, u shares the same property. Thus u is the maximal solution.  We give below a sufficient condition for the existence of at least one solution of .5.1.1/ in a domain. Proposition 5.1.2. Let  be a Greenian domain1 in RN and g 2 GQ0 is such that g.x, / is non-decreasing. If there exists some r0 2 R such that Z ess sup G .x, y/g.x, r0/dx < 1, (5.1.3) y2K



for every compact subset K  , where G is the Green kernel in , there exists a solution of .5.1.1/ in . Proof. By assumption, w D GŒg.., r0/ is locally bounded in . Since g.., x/ is monotone, .5.1.3/ holds if r0 is replaced by r  r0. Then v D r0  w is a weak solution of v C g.x, r0 / D 0 in . Furthermore v  r0, which implies that g ı v 2 L1loc ./ and v is a subsolution of (5.1.1). The constant function r0 is a supersolution and r 7! g.x, r / is non-decreasing, therefore there exists a solution u  of .5.1.1/ and v  u  r0. Remark 5.1. If the nonlinearity is too singular near @, it may happen that no solution at all exists. This is the case for the problem u C .1  jxj/2 e u D 0 in B1 .

(5.1.4)

c .x0 / of As an example consider the case where the domain is the complement BR the ball BR .x0/. Then the Green kernel is given by (see e.g. [5, p. 96]) 8  jyx j jxy  j  0 ˆ if N D 2 < ln R jxyj c (5.1.5) GBR .x0/ .x, y/ D   ˆ : jx  yj2N  jyx0jjxy j 2N if N  3. R

Corollary 5.1.3. Let  be a proper exterior domain in RN with a smooth boundary c  . Assume that g 2 GQ0 is such that g.x, / is non-decreasing and that such that BR c there exist r0 and y 2 B RC1 such that Z Z g.x, r0/.x/dx C g.x, r0/GBRc .x, y/dx < 1, (5.1.6) \BRC1

c BRC1

where .x/ D @ .x/ :D dist .x, @). Then .5.1.1/ admits a maximal solution. 1

A domain is Greenian if there exists a Green function. If N  3 every domain is Greenian, if N D 2 a domain is Greenian if and only if its complement has zero Newtonian capacity.

164

Chapter 5 Classical theory of maximal and large solutions c

c

Proof. There exists R0 < R such that B R    B R0 . By .5.1.5/ there exists a constant c > 0 such that GBRc .x, y/  GB c 0 .x, y/  cGBRc .x, y/ R

c 8.x, y/ 2 BRC1 .

(5.1.7)

Furthermore G .x, y/  GB c 0 .x, y/, for all x and y in . Since @ is smooth, R c G .x, y/  C.x/ for all x 2  \ BRC1 and y 2 BRC1 . Then .5.1.6/ implies .5.1.3/. The conclusion follows from Theorem 5.1.1 and Proposition 5.1.2.  If g does not depend on x, the previous results can be made more precise: Corollary 5.1.4. Let  be a proper exterior domain in RN with a smooth boundary c such that BR  . Assume that g 2 C.R/ is non-decreasing, positive, convex and satisfies the extended Keller–Osserman condition. (i) If N  3 there exists no solution of .5.1.1/ in . (ii) If N D 2 there exists a solution of .5.1.1/ in  if and only if there exists ˛ < 0 such that Z 1 g.˛t /e 2t dt < 1.

(5.1.8)

0 c   and that there exists a solution u of .5.1.1/ in . If u is the Proof. Assume BR spherical average of u, it satisfies

u00 C

N 1 0 u  g.u/ r

in ŒR, 1/.

(5.1.9)

If N D 2 we set .t / D u.r / with r D jxj and t D ln r 2 .T , 1/. Then

00 .t /  e 2t g ı .t /

in ŒT , 1/.

Therefore the function is convex and there exists ˛ < 0 such that .t /  ˛t . Since g is non-decreasing, Z t 0 0

.t /  .T / C e 2s g.˛s/ds. T

This implies that .5.1.8/ must hold, otherwise lim t !1 0 .t / D 1, which contradicts Theorem 4.1.2. Conversely, if we assume that .5.1.8/ holds for some ˛ < 0, for any m > T there exists a unique solution D m to

00 .t / D e 2t g. .t //

in .T , m/,

.T / D .m/ D 0.

Furthermore m 7! m .t / is decreasing. Because of .5.1.8/, the function Z t 1 .t  s/e 2s g.˛.s  T //ds H.t / :D t T T

165

Section 5.1 Maximal solutions

is uniformly bounded on .T , 1/. Set b D sup¹H.t / : t > T º and a D ˛  b. Then Z t .t  s/e 2s g.˛.s  T //ds  ˛.t  T / 8t  T . a.t  T / C T

If we define Q / D a.t  T / C

.t

Z

t T

.t  s/e 2s g.˛.s  T //ds,

Q / D 0 and Q 00 .t / D e 2t g.˛.t  T //  g. .t Q //. By the maximum then Q  0, .T principle Q  m on ŒT , m. Let :D limm!1 m . Then u.jxj/ D .e jxj/ is a solution of .5.1.1/ in BecT . Next, if N  3 we set .s/ D su.r / with s D 00

.s/  C.N /s

N=.N 2/2

r N 2 N 2

2 ŒS, 1/. Then

g. .s/=s/

in ŒS, 1/,

for some C.N / > 0. Since is convex, .s/=s  ı for some ı > 0. Then g. .s/=s/  g.ı/, and the above inequality implies lim inf s N=.N 2/ .s/ > 0. s!1

This leads again to lim supjxj!1 u.x/ D 1.



When g.., 0/ D 0 there always exists a subsolution. Theorem 5.1.5. Let  be a proper domain in RN and g 2 GQ0. If g satisfies the positive Keller–Osserman condition, g.x, 0/ D 0 and g.x, ./ is non-decreasing, then .5.1.1/ admits a maximal solution in . Furthermore, if  is an exterior domain, it satisfies N D 0. lim sup u.x/

(5.1.10)

jxj!1

Proof. The construction is similar to the one of Theorem 5.1.1, the only difference lies in the upper estimate of the un in n which becomes un .x/  fQ .n .x// (we recall that the function f  introduced in Theorem 4.1.4 satisfies lim!0 f  ./ D 1 and lim!1 f  ./ D 0).  A variant of the previous result is the following: Theorem 5.1.6. Let  be a proper domain in RN and g 2 GQ0 is such that g.x, ./ is non-decreasing. If g and g  satisfy the positive Keller–Osserman condition, then .5.1.1/ admits a maximal solution and a minimal solution in . Proof. Notice that the assumptions imply g  2 GQ0 and g.x, 0/ D 0. Since g satisfies the positive Keller–Osserman condition, there exists a maximal solution u by Theorem 5.1.5, and u  0. Similarly, if g  satisfies the positive Keller–Osserman condition, there exists a maximal solution u  0 to u C g  ı u D 0 in . Clearly, u D u is the minimal solution of .5.1.1/ in .

(5.1.11) 

166

Chapter 5 Classical theory of maximal and large solutions

5.1.2 Local conditions When g.x, r / depends on x, we have introduced a local Keller–Osserman condition version in Definition 3.1.9. Similarly, the local extended Keller–Osserman condition stipulates that the estimate g.x, r /  h0 > 0 holds on 0 R (up to a negligible set) 0 where 0 is an open subset R t such that  b , .3.1.14/ holds for every a 2 R, with H replaced by H0 .t / D 0 h0 .s/ds. This does not alter the existence of a maximal solution. Theorem 5.1.7. Let  be a proper domain in RN and g 2 GQe satisfies the local extended Keller–Osserman condition. Assume also that g.x, ./ is non-decreasing. If .5.1.1/ admits a solution in , it admits a maximal solution. Proof. The proof of the result is an adaptation of the one of Theorem 5.1.1. We consider an increasing sequence ¹n º of relatively compact domains such that n b nC1 and  D [n ; for each n we denote by un the solution of u C g ı u D 0 lim u D 1 n.x/!0

in n on @n .

(5.1.12)

It verifies un .x/  fgn .n .x// 8x 2 n

(5.1.13)

where the fgn is the function constructed in Theorem 4.1.2 relative to n . Since un is bounded from below by the restriction to  of any solution defined in , the  sequence ¹un º is decreasing. The remainder of the proof is standard. Similar results can be easily stated when the positive Keller–Osserman condition is replaced by the local positive Keller–Osserman condition. This condition is that for any open domain 0 b , there exists h0  0 such that g.x, r /  h0 .r / in 0 R, and there exists a0  0 such that (3.1.14) holds with a D a0 . Theorem 5.1.8. Let  be a proper domain in RN and g 2 GQ0 is such that g.x, ./ is non-decreasing. If g and g  satisfy the local positive Keller–Osserman condition, then .5.1.1/ admits a maximal and a minimal solution in .

5.2 Large solutions 5.2.1 General nonlinearities An important question concerning maximal solutions of .5.1.1/ in   RN is whether they tend to infinity on the boundary.

167

Section 5.2 Large solutions

Definition 5.2.1. A solution u of .5.1.1/ in  is called a large solution if for any compact set K  , (5.2.1) lim u.x/ D 1. x2K\ .x/ ! 0

This statement is equivalent to lim u.x/ D 1

x!y

8y 2 @.

(5.2.2)

Definition 5.2.2. A domain   RN satisfies the Wiener condition or is a Wiener domain if, for every a 2 @, there holds Z

1 0

  ds C1,2 Bs .a/ \ c N 1 D 1, s

(5.2.3)

where C1,2 denotes the Newtonian capacity. If this condition holds and  is bounded, the Dirichlet problem with continuous boundary data admits a solution in C./ with boundary value [53, Chap 2]. Theorem 5.2.3. Assume that   RN is a Wiener domain and g 2 C. R/ is such that g.x, ./ is non-decreasing. If g 2 GQ0 satisfies the positive Keller–Osserman condi./ of .5.1.1/, then the maximal solution u tion and there exists a solution u0 2 L1 loc is a large solution. Proof. For R, k > 0 we set R D  \ BR and denote by u D uR,k the solution of u C g ı u D 0 uDk u D u0

in R on @ \ BR on  \ @BR .

(5.2.4)

If we take k > u0 , u0  uR,k  u by the maximum principle. When k ! 1, uR,k " uR,1 which satisfies (5.2.4) as uR,k except that it tends to 1 when x ! y 2 @\BR0 for any 0 < R0 < R. Furthermore u0  uR,1  u (resp. 0  uR,1  u) if g 2 GQe (resp. g 2 GQ0 ). This implies that R 7! uR,1 is increasing and limx!y u.x/ D 1, for  any y 2 @ \ BR and any R > 0. When there exist solutions with isolated singularities, we show that the maximal solution of u C g ı u D 0 (5.2.5) in  is a large solution without any regularity assumption on @, but g needs to satisfy an extra condition for that.

168

Chapter 5 Classical theory of maximal and large solutions

Lemma 5.2.4. Assume that N  3 and g 2 C.R/ is non-decreasing with g.0/  0 and satisfies the weak singularity assumption. Then for any R > 0, ˇ 2 R and k > 0 there exists a unique v D vk,R,ˇ continuous on .0, R which satisfies v 00 

N 1 0 v Cgıv D 0 r v.R/ D ˇ N 2 v.r / D k. lim r

on .0, R/ (5.2.6)

r !0

Furthermore .k, ˇ/ 7! vk,R,ˇ is non-decreasing. Proof. Notice that the assumption on g is equivalent to Z 1 g.t /t 2.N 1/=.N 2/ dt < 1

(5.2.7)

1

The change of variables s D r N 2 =.N  2/ and .s/ D sv.r / transforms .5.2.6/ into

00 D C.N /s N=.N 2/2 g. =s/

.S/ D ˇ  :D Sˇ

.0/ D k  :D k=.N  2/.

on .0, S/ (5.2.8)

where S D RN 2 =.N  2/. For 2 .0, 1/, let  be the solution of

00 D C.N /.s C /N=.N 2/2 g. =.s C //

.S/ D ˇ  :D Sˇ

.0/ D k  :D k=.N  2/.

on .0, S/ (5.2.9)

Since g is non-negative,  is convex on Œ0, S. If  is non-decreasing, min¹ˇ , k  º    max¹ˇ  , k  º. Since 00 is bounded on ŒS=2, S, it implies that 0 remains bounded on this interval. If  is not monotone, there exists s0 2 .0, S/ such that 0 .s0 / D 0. Then  is decreasing on .0, s0 /, increasing on .s0 , S/ and there holds Z s0 0 0   .s/  C.N / . C /N=.N 2/2 g.k  =. C //d 8s 2 .0, s0 /. s

Thus

 .s0 /  k   C.N /  k   C.N / But Z s0C 

(5.2.10) R s0C R s0C 

R s0C 

s

 N=.N 2/2 g.k  =/dds

 N=.N 2/2 .  /g.k  =/d.

 N=.N 2/2 .  /g.k  =/d D k 

2 N 2

Z

k  k s0 C

N 1

(5.2.11)

t 2 N 2 .k   t /g.t /dt .

169

Section 5.2 Large solutions

By .5.2.7/, this last integral is bounded independently of 2 .0, 1. Therefore  remains bounded on Œ0, S, and since 00 is bounded on ŒS=2, S, the same boundedness property holds for 0 . Integrating .5.2.9/ we obtain Z S 0 0 . C /N=.N 2/2 g.  ./=. C //d,

 .s/ D  .S/  C.N / s

and, for 0  s 

s0

 S,

ˇ ˇ ˇ  .s/   .s 0 /ˇ  C.s 0 s/CC.N /

Z sZ0 s

S t

. C /N=.N 2/2 g.  ./=. C //ddt .

For c > 0, set Z ˆc .s/ D Then

SC1Z SC1 s

Z

SC1Z SC1

lim ˆc .s/ D

s!0

 N=.N 2/2 g.c=/ddt

t

0

t

8s 2 .0, S C 1.

 N=.N 2/2 g.c=/ddt D

Z

SC1

(5.2.12)

 2=.N 2/ g.c=/d.

0

By .5.2.7/ this last integral exists. Therefore ˆc is uniformly continuous on .0, S C 1 and ˇ ˇ ˇ ˇ ˇ  .s/   .s 0 /ˇ  C.s 0  s/ C C.N / ˇˆc .s 0 /  ˆc .s/ˇ 8s 2 Œ0, S, with c D max¹ˇ  , k  º. Then the family of functions ¹  º is equicontinuous on Œ0, S and there exists n and 2 C.Œ0, S/ such that n ! uniformly in Œ0, S. This implies that is a solution of .5.2.8/. Uniqueness follows from the fact that if and

Q are two solutions of the equation, s 7! . .s/  Q .s//2 is a convex function.  Lemma 5.2.5. Assume N D 2, g 2 C.R/ with g.0/  0 is non-decreasing with finite exponential order of growth agC (see the definition in (4.5.14)). Then for any R > 0, ˇ 2 R and 0 < k < 2=agC there exists a unique v D vk 2 C..0, R/ which satisfies 1 v 00  v 0 C g ı v D 0 on .0, R/ r v.R/ D ˇ lim v.r /= ln r D k,

(5.2.13)

r !0

and again k 7! vk is non-decreasing. Proof. For > 0, let 

00

D



be the solution of

C .t C /3 e  tC g. =.t C // D 0 on .0, 1/ .1/ D ˇ lim .t / D k, 2

t !0

(5.2.14)

170

Chapter 5 Classical theory of maximal and large solutions

where D R2 e 2. Since  is convex,   max.ˇ, k/. If  is monotone there also holds   min¹ˇ, kº and we derive that 0 remains bounded on Œ1=2, 1. If  is not monotone, there exists t0 2 .0, 1/ where  achieves it minimum. Therefore Z t0 Z t0 2 .s C /3 e  sC g.  .s/=.s C //dsdt k   .t0/ D Z

0 t0



Z

 k

t0 t

0

Z

t

k  k t0 C

.s C /3 e  sC g.k=.s C //dsdt 2

    2 e k g./dt . 1 k

Since 2=k > agC , this last integral is bounded independently of . Therefore  remains uniformly bounded on .0, 1/. This implies that 0 remains uniformly bounded on Œ1=2, 1/. For 0 < t < t 0 < 1 and using  ./  .ˇ  k/ C k, we obtain Z tZ0 1 2 . C /3 e  C g.  ./=. C //d ds j  .t /   .t 0 /j  C.t 0  t / C t

 C.t 0  t / C

Z tZ0 t

Z tZ0

s

1 s

. C /3 e  C g 2

 k C .ˇ  k/   C

d ds

 k  .ˇ  k/ C ˇ  k d ds.  t sC (5.2.15) We choose < 1 such that 0 < k 0 :D k  .ˇ  k/ < 2=agC and set   0 Z 2Z 2 k 3  2 ‰.x/ D C ˇ  k d ds.  e g  x s  C.t 0  t / C

Then

ˇ ˇ

 .t /



 .t

0

1C

 3 e   g 2

ˇ ˇ ˇ /ˇ  C jt 0  t j C ˇ‰.t /  ‰.t 0 /ˇ .

Furthermore  0  0   Z 2Z 2 Z 2 k k 3  2 2  2  e g  e g C ˇ  k d ds D C ˇ  k d   0 s 0 Z 1 2.ˇk/ 2 e  k0 g ./ d , D k 01 e k0 0 ˇ kC k2

and this last integral is finite since agC < 2=k 0 . Consequently ‰ is uniformly continuous on .0, 2. This implies that the set of functions ¹  º is equicontinuous and there exists a solution of 

00

C t 3 e  t g. =t / D 0 on .0, 1/ .1/ D ˇ lim .t / D k. 2

t !0

(5.2.16)

171

Section 5.2 Large solutions

  1      1 Re 1 ln D t If we set u.x/ D ln Re .t / with t D ln Re , then jxj jxj jxj u satisfies .5.2.13/. Uniqueness follows by convexity.  Remark 5.2. If, in Lemma 5.2.4 and Lemma 5.2.5, it is assumed moreover that g satisfies the positive Keller–Osserman condition, there exists v1 D limk!1 vk . It satisfies the equation in .0, R/, achieves the value ˇ for r D R and .i/ .ii/

lim r N 2 v1 .r / D 1

if N  3

lim v1 .r /= ln r D 1

if N D 2.

r !0

r !0

(5.2.17)

Theorem 5.2.6. Let   RN , N  2. Assume g 2 C.R/ is non-decreasing, nonnegative on RC and g satisfies the positive Keller–Osserman condition. Assume also either N  3 and the weak singularity assumption holds, or N D 2 and agC < 1. If .5.2.5/ admits a maximal solution u in , it is a large solution. Proof. Let ¹n º be an increasing sequence of relatively compact smooth open subsets of  such that [n D , and, for each integer n, let u D un be the solution of u C g ı u D 0 in n lim u.x/ D 1.

(5.2.18)

@n .x/!0

Then un # u when n ! 1. If y 2 @ and R > 0, we set ˇ D ˇ.R, y/ :D min¹u.z/ : z 2  \ @BR .y/º. Then un .x/  v1,R,ˇ .jx  yj/ in n \ BR .y/ (see Lemma 5.2.4 for the definition of vk,R,ˇ ). Letting n ! 1 we obtain u.x/  v1,R,ˇ .jx  yj/ in  \ BR .y/. Since for any x 2 , there exists yx 2 @ such that .x/ D jx  yx j, we derive u.x/  v1,.x/,ˇ.x/..x//

8x 2  \ B.x/.yx /,

with ˇ.x/ D min¹u.z/ : z 2  \ @B.x/.yx /º. This implies the claim.

(5.2.19) 

5.2.2 The power and exponential cases For q > 1 and h 2 C./, h > 0, we consider u C h.x/jujq1 u D 0

(5.2.20)

in an open domain   RN . Then there exist a maximal solution u and a minimal solution u.

172

Chapter 5 Classical theory of maximal and large solutions

Proposition 5.2.7. Any solution of .5.2.20/ in  satisfies ju.x/j 

C.N , q/

(5.2.21)

1

.D.x// q1

where D.x/ D

sup

inf

0
r 2h.y/.

(5.2.22)

Proof. Let x 2  and 0 < r < .x/. If A D A.r , x/ D inf¹h.y/ : y 2 Br .x/º, u C Ajujq1 u  0 in Br .x/.

(5.2.23)

By Keller–Osserman estimate, C.N , q/ u.x/    1 . Ar 2 q1 Optimizing over r and replacing u by u yields .5.2.21/.

(5.2.24) 

An important case corresponds to the case where h depends only on .x/. Corollary 5.2.8. Assume that h.x/ D `..x//, where ` is a continuous positive function defined on .0, 1/. ® ¯ (i) If ` is decreasing D.x/ D sup r 2h..x/ C r / : 0 < r < .x/ . ¯ ® (ii) If ` is increasing D.x/ D sup r 2h..x/  r / : 0 < r < .x/ . Remark 5.3. The question whether the maximal solution is a large solution is difficult to answer since two obstacles exist. The first one is due to the interaction between the regularity of the boundary and the power q: only the case h 1 is now completely solved and its treatment requires the use of Bessel capacities. The second one is that obstruction to the existence of a large solution may come from the term h.x/. Theorem 5.2.9. Assume that  is a bounded smooth domain. If q > 1, h 2 L1 .; / and then .5.2.20/ admits a solution U which satisfies lim U.x/ D 1

x!a

for almost all a 2 @

(5.2.25)

where the limit is taken non-tangentially and the almost statement is understood in the sense of the HN 1 Hausdorff measure. Furthermore, if h.x/ D `..x//, U is a large solution. Proof. Assume h 2 L1.; /. For m, k 2 N we set hm D min¹m, h.x/º and denote by u D um,k the solution of u C hm .x/uq D 0 in  u D k on @.

(5.2.26)

173

Section 5.2 Large solutions

By the maximum principle .m, k/ 7! um,k is decreasing with respect to m and increasing with respect to k. Furthermore um,k  k. Therefore um,k  wm,k where wm,k D k  k q G Œhm . Since h 2 L1.; /, limm!1 G Œhm  D G Œh. When m ! 1 um,k converges to some uk which satisfies uk  k  k q G Œh and

q

uk C h.x/uk D 0 in  uk D k in @.

(5.2.27)

By Fatou’s theorem, for HN 1 -almost all a 2 @, uk .x/ ! k when x ! a, non-tangentially. When k " 1, uk " U which is a solution of .5.2.20/ in  and satisfies .5.2.25/. If h D ` ı  then lim.x/!0 G Œh.x/ D 0, which implies that lim.x/!0 uk .x/ D k. Letting k ! 1 yields lim.x/!0 U.x/ D 1.  Next we consider the case where h.x/ D ..x//˛ . The following result is a consequence of Corollary 5.2.8 and Theorem 5.2.10. Theorem 5.2.10. Let  be a bounded smooth domain, q > 1 and ˛ 2 R. Then equation (5.2.28) u C  ˛ jujq1 u D 0 admits a large solution in  if and only if ˛ > 2. When @ is not smooth, existence of large solution holds in the subcritical case. Theorem 5.2.11. Let   RN be a proper domain. Then for any ˛  0 and 1 < q < qe,˛ D .N C ˛/=.N  2/ if N  3, or q > 1 if N D 2, equation .5.2.28/ admits a large solution in . Proof. Since 1 < q < .N C ˛/=.N  2/, it follows from Theorem 4.2.6 that there exists a (unique) function u D u1 satisfying in B1 u C jxj˛ uq D 0 uD0 on @B1 1=.q1/ lim jxj.2C˛/=.q1/ u.x/ D N ,q,˛ .

(5.2.29)

x!0

Since ˛  0, for any y 2 @, the function u1 ..  y/ satisfies u C  ˛ uq  0

in B1 .y/.

The maximal solution u is the decreasing limit of the sequence ¹un º of solutions of u C  ˛ jujq1 u D 0 uD1

in n on @n ,

(5.2.30)

174

Chapter 5 Classical theory of maximal and large solutions

where ¹n º is a smooth exhaustion of . Applying the maximum principle in n \ B1.y/ and letting n ! 1 yields u.x/  u1 .x  y/

8x 2  \ B1 .y/.

(5.2.31)

This implies the claim. Next, for a > 0 and h 2 C./, h > 0, we consider



u C h.x/e au D 0.

(5.2.32)

Then g.x, r / D h.x/e ar 2 GQe and it satisfies the extended Keller–Osserman condition. However there may not exist any maximal solution, even in the case where h D 1. Proposition 5.2.12. Any solution of .5.2.32/ in  satisfies   1 1 3 u.x/  ln  ln a C N a D.x/ 2

(5.2.33)

where D.x/ D

sup

inf

0
r 2h.y/.

(5.2.34)

Theorem 5.2.9 is valid for equation .5.2.32/ without any real modification of the proof. Similarly, Theorem 5.2.10 holds if equation .5.2.28/ is replaced by u C  ˛ e au D 0.

(5.2.35)

Theorem 5.2.13. Let   R2 be a proper domain and assume a > 0 and ˛  0. If equation .5.2.35/ admits a solution in , it admits a large solution. Proof. If u is a solution of u C jxj˛ e au D 0 uD0 then v.x/ D u.x/ C

˛ a

in B1 n ¹0º on @B1 ,

(5.2.36)

ln jxj satisfies v C e av D 0 vD0

in B1 n ¹0º on @B1 .

Using Lemma 5.2.5 we derive that for any 0 < k < .2 C ˛/=a there exists a unique solution uk of .5.2.36/ such that lim uk .x/= ln.1=jxj/ D k.

x!0

(5.2.37)

If y 2 @, the function x 7! uk .x  y/ is a subsolution of .5.2.35/ in B1 .y/. As in the proof of Theorem 5.2.11, it implies un  uk ..  y/, then u  uk ..  y/ and the claim follows. 

175

Section 5.3 Uniqueness of large solutions

5.3 Uniqueness of large solutions The first result for proving uniqueness of large solutions of u C jujq1 u D 0

(5.3.1)

(q > 1), was obtained in proving that, if the domain is C 2 , there holds  u.x/ D

2.q C 1/ .q  1/2



1 q1

..x// q1 .1 C ı.1// 2

as .x/ ! 0.

(5.3.2)

The regularity of the boundary enables to consider inner or outer balls tangent to the boundary and to construct radial solutions of (5.3.1) therein. Since the first term of the asymptotic expansion of a radial large solution can be obtained by ODE techniques, (5.3.2) follows by comparison. As a consequence, if u and u0 are two large solutions, then for any > 0, .1 C /u is a supersolution which dominates strictly u0 in a neighborhood of @. By the maximum principle u0  .1 C /u in . Letting ! 0 yields u0  u. Similarly u  u0 . A more direct method that we shall develop below and does not need any regularity for @, is to prove directly that two large solutions u and u0 of .5.3.1/ verify u.x/ D 1. (5.3.3) lim .x/!0 u0 .x/ Finally, we prove that uniqueness holds if there exist a constant C  1 and ı > 0 such that any couple .u, u0 / of large solutions is subject to the estimate C 1 u.x/  u0 .x/  C u.x/

8x 2  s.t. .x/  ı.

(5.3.4)

In all these methods the convexity of the nonlinear term is crucial.

5.3.1 General uniqueness results We recall the following classical result which exists in various forms (see e.g. [95, 101]). Proposition 5.3.1. Let   RN be a domain, L a second order elliptic operator with smooth coefficients and h, h , h 2 C. R/. We assume that h D h C h, where r 7! h .x, r / is non-decreasing for every x 2 , and .x, r / 7! h.x, r / is locally Lipschitz continuous with respect to the r variable, uniformly when the x variable stays in a compact subset of . Let u and u be two functions belonging to 1,2 C./ \ Wloc ./ and satisfying .i/ .ii/ .iii/

Lu  h.x, u /  0 Lu  h.x, u /  0 u  u

in  in  in ,

(5.3.5)

176

Chapter 5 Classical theory of maximal and large solutions

where the equations are understood in a weak sense, then there exists a C 1 ./ function u which satisfies .i/ .ii/

Lu  h.x, u/ D 0 u  u  u

in , in .

(5.3.6)

Theorem 5.3.2. Let   RN be a proper domain. Let g 2 C. R/ such that g.x, ./ is non-decreasing and convex. Assume also that equation u C g ı u D 0

(5.3.7)

admits at least one bounded solution v in . If there exist a maximal solution u of .5.3.7/ and two constants K > 1 and ı > 0, depending on  and g, such that 0  u.x/  Ku.x/ ,

8x 2  s.t. .x/  ı,

(5.3.8)

for any large solutions u in , then u is the unique large solution of .5.3.7/. Proof. By the maximum principle, a large solution is bounded from below by v in . If u ¤ u, we put 1 .u  u/. w Du 2K Then w > 0 by the strict maximum principle and       1 u 1 1 w C g.x, w/ D g x, 1 C u  1C gıuC g.x, u/. 2K 2K 2K 2K    u u 1 2K 1C u C D u, 1 C 2K 2K 2K 1 C 2K by convexity assumption     1 1 2K u g x, 1 C u C g.x, u/  g ı u. 1 C 2K 2K 2K 1 C 2K Since

Therefore w is a supersolution in  and it is larger than .K C 1/u=2K by .5.3.8/. For  2 .0, 1/, w D u C .1  /v is a subsolution. If we fix  D .K C 1/=2K, then w < w in a neighborhood of @ and therefore in  because g.x, ./ is non-decreasing. By Proposition 5.3.1, there exists a solution u1 of .5.3.7/ in  which satisfies w  u1  w. Therefore u1 is a large solution, thus it satisfies .5.3.8/. Notice also that   1 1 .u  u/ H) u  u1  1 C .u  u/. (5.3.9) u1  w D u  2K 2K Replacing u by u1 , which is a large solution, we introduce w1 D u1 

1 .u  u1 / and w1, D u1 C .1  /v, 2K

177

Section 5.3 Uniqueness of large solutions

where 0 <  < .K C 1/=2K. Again by convexity w1 and w1, are respectively a super- and a subsolution of .5.3.8/, and since w1, < w1 , there exists a solution u2 of .5.3.7/ in  which satisfies w1,  u2  w1 . Then u2 is a large solution and   1 1 .u  u1 / H) u  u2  1 C .u  u1 /. u2  u1  (5.3.10) 2K 2K Iterating this process, we construct a sequence of large solutions ¹un º of .5.3.7/ in  with u0 D u, which satisfy .5.3.8/, are larger than v and for which   1 1 un  un1  .u  un1 / H) u  un  1 C .u  un1 /. (5.3.11) 2K 2K Consequently, since un  v there holds   1 n u  v  u  un  1 C .u  u/ in . 2K We derive a contradiction by letting n ! 1.

(5.3.12) 

Theorem 5.3.3. Let  be a bounded domain in RN such that @ is locally a continuous graph. Assume that g 2 C.R/ with g.0/  0 is convex, non-decreasing and gC satisfies the Keller–Osserman condition. Then equation u C g ı u D 0

(5.3.13)

admits at most one large solution in . Proof. Since g is non-decreasing and convex it satisfies g.a C b/  g.a/ C g.b/  L

8a, b  0,

(5.3.14)

where L D g.0/  0. Let R > 0 such that   BR . Any large solution of (5.3.13) is bounded from below by the solution w of the same equation with zero boundary data. Since @ is locally the graph of a continuous function, for every boundary point P there exist an open neighborhood QP , a set of coordinates  D .1 , : : : , N / obtained from x by rotation, and a function FP 2 C.RN 1 / such that QP \  D QP \ G.FP / with G.FP / D ¹ : N < FP .1 , : : : , N 1 /º. (5.3.15) We can assume that QP is a bounded cylindrical domain centered at P , with axis parallel to the N -axis: ˇ ˇ QP D ¹ D .0 , N /, ˇ0 ˇ < RP , jN j < TP º, (5.3.16) where  D   P and 0 D .1 , : : : , N 1 /. We can also suppose that @ is bounded away from the top and the bottom of the cylinder QP and that @\ QN P D @ \ QP . We denote N (5.3.17) ‚ D  \ QP , 1 D QP \ @, 2 D @QP \ . N We can also assume that QP  BR for any P .

178

Chapter 5 Classical theory of maximal and large solutions

Step 1. Assume that there exists a large solution u. We claim that there exists a positive function v 2 C 2 .‚/ solution of .5.3.13/ in ‚, with the properties .i/ .ii/

v.x/ ! 1 v.x/ ! 0

locally uniformly as x ! 1 , locally uniformly as x ! 2 ,

(5.3.18)

where “v.x/ ! 0 (or 1) locally uniformly as x ! i ” means that, for every compact subset K contained in the relative interior of i , there holds v.x/ ! 0 (or 1) as dist .x, K/ ! 0. Set FQP .0 / D FP .0 C P 0 / and let ¹fnº be an increasing sequence of smooth positive functions defined in the closed .N  1/-ball DP D ¹0 D .1, : : : , N 1 / : j0 j  RP º, which converges uniformly in DP to FQP . We set ‚n D ¹ : j0 j < RP , TP < N < fn .0 /º, 2,n D @QP \ @‚n and 1,n D @‚n n@QP . Let vn,k 2 C.‚n / be a positive solution of .5.3.13/ in ‚n such that vn,k ./ D k for  2 1,n , vn,k .0 , N / D 0 for  2 2,n s.t.  TP  N  fn .0 /  1=n, (5.3.19) vn,k .0 , N / D n.N  fn .0 / C 1=n/k ˇ ˇ for ˇ0 ˇ D RP , fn .0 /  1=n  N  fn .0 /. For fixed n, the sequence ¹vn,k º is increasing with respect to k. It converges, as k " 1, to some function vn 2 C.‚n /, solution of .5.3.13/ in ‚n . Moreover .i/ .ii/

vn .x/ ! 1 vn .x/ ! 0

locally uniformly as x ! 1,n ,  locally uniformly as x ! 2,n ,

 D ¹ 2  0 where 2,n 2,n : N  fn . /  1=nº. The sequence ¹vn º is decreasing and converges locally uniformly in ‚ to some continuous function v which satisfies .5.3.13/ in ‚. Therefore v satisfies (5.3.18) (ii). Moreover, v is bounded from below by ` which is the minimal value of the function v  solution of

v  C g.v  / D 0 

vQ D 0

in QP ,

(5.3.20)

on @QP .

For proving that v satisfies (5.3.18) (i), we introduce a large solution w of .5.3.13/ in QP (which exists since @QP is Lipschitz continuous, and is bounded from below by some `Q since QN P  BR ). We put Q 0º, ` D min¹` , `,

v` D v  `,

vn,` D vn  `,

w` D w  `.

Then vn,` and w` are non-negative and they satisfy .vn,` C w` / C g.vn,` C w` / D g.vn,` C w` /  g.vn,` C `/  g.w` C `/  L (5.3.21)

179

Section 5.3 Uniqueness of large solutions

in ‚n . The last inequality follows from (5.3.14); note that, by definition, `  0. Let Q be the solution of Q D L Q D 0

in BR ,

(5.3.22)

on @BR .

Since Q is non-negative, the function Wn D vn,` C w` C Q satisfies Wn C g.Wn /  0 in ‚n

(5.3.23)

and dominates u on @‚n. Therefore Wn  u in ‚n . Letting n ! 1 implies v` C w` C Q  u in ‚; then v satisfies (5.3.18) (i). We can also move the domain ‚ in the outward direction N > 0 by defining, for > 0 small enough, ‚ D ¹.0, N / : .0 , N  / 2 ‚º. We can choose small enough so that ‚  BR . The shifted function v , defined in ‚ satisfies (5.3.18) on the corresponding lateral and upper boundaries of ‚ . Since g is non-decreasing, v  `QC is a subsolution of (5.3.13) in ‚ \  and is smaller than u on the boundary. Thus u  v  `QC . Letting ! 0 we finally obtain v  `QC  u  v C w C Q  2` in ‚.

(5.3.24)

Step 2. Let uN be the maximal solution and u any large solution of .5.3.13/ in . Since @ is compact, it follows from (5.3.24), that uN D 1. .x/!0 u lim

(5.3.25)

If assumption (i) holds, the function 2u satisfies .2u/ C g.2u/  L, thus .2u C / Q C g.2u C / Q  0. Applying (5.3.25) and the maximum principle (since .uN  2u  / Q C has compact support in ), we infer that uN  2u C Q  3u in . Therefore estimate .5.3.8/ holds, and the proof follows from Theorem 5.3.2. If assumption (ii) holds, for any > 0, the function u D .1 C /u satisfies u C g.u /   L. Thus Q C g.u C / Q  0. .u C /

180

Chapter 5 Classical theory of maximal and large solutions

Applying again (5.3.25) we obtain u C Q  u. N Letting ! 0 leads to

u  u, N 

and uniqueness follows.

If @ is locally a continuous graph, there exists a decreasing sequence of smooth domains with a compact boundary such that their intersection contains . Corollary 5.3.4. Assume that  and g satisfy the assumptions of Theorem 5.3.3, and (5.3.13) admits a large solution u in  (necessarily unique). Let ¹0n º be a decreasing sequence of smooth bounded domains containing  and such that \n 0n D . If u0n is the minimal large solution of (5.3.13) in 0n , then sup u0n D lim u0n D u.

n0

(5.3.26)

n!1

Proof. We follow the approach of Theorem 5.3.3 except that we regularize the local approximations of the boundary. With the same notation, for every P 2 @, let QP be defined by (5.3.16) and ‚, 1 and 2 by (5.3.17). For n 2 N there exists FP ,n 2 C 1 .B RP .P // such that FP . 0 /  n1 < FP ,n . 0 / < FP . 0 / 8 0 s.t. j 0 j < RP , we assume that n1 < TP =3. Accordingly, we define G.FP ,n / D ¹ : N < FP ,n . 0 /º, ‚n D  \ G.FP ,n / D  \ ¹ : N < FP ,n . 0 /º and 1,n D QP \ ¹ : N D FP ,n . 0 /º, 2,n D @QP \ G.FP ,n /. We denote by u D uP ,n the large solution of (5.3.13) in ‚n (unique by Theorem 5.3.3), and by vP ,1,n and vP ,2,n two solutions of (5.3.13) in ‚n satisfying

and

vP ,1,n .x/ ! 1 vP ,1,n .x/ ! 0

locally uniformly as x ! 1,n locally uniformly as x ! 2,n

(5.3.27)

vP ,2,n .x/ ! 0 vP ,2,n .x/ ! 1

locally uniformly as x ! 1,n locally uniformly as x ! 2,n .

(5.3.28)

Because of uniqueness, max.vP ,1,n , vP ,2,n /  uP ,n  vP ,1,n C vP ,2,n

in ‚n.

(5.3.29)

181

Section 5.3 Uniqueness of large solutions

Let ¹n º be an increasing sequence of smooth domains such that [n n D  and dH .@n , @/ ! 0 as n ! 1 (dH .., ./ is the Hausdorff distance). We denote by un the minimal large solution of (5.3.13) in n (obtained as the increasing limit of solutions with boundary data k ! 1). Then, for any n 2 N , there exists mn 2 N such that for m  mn there holds ‚n  m and um  uP ,n

in ‚n .

(5.3.30)

For every m 2 N , there exists nm 2 N such that for n  nm there holds vP ,1,n  um

in ‚n \ m .

(5.3.31)

Similarly, we define a minimal large solution u0m of (5.3.13) in 0n , and we move the set ‚n in the direction N and set ‚0n :D ¹ :   2n1 eN 2 ‚n º. Accordingly, 0 ,  0 , v0 0 we define FP0 ,n D FP ,n C 2n1 , 1,n 2,n P ,1,n and vP ,2,n which correspond to the same objects up to the translation 2n1 eN . Thus for any n 2 N , there exists mn 2 N such that for m  mn there holds ‚n  m and vP0 ,1,n  u0m

in ‚0n \ 0m ,

(5.3.32)

and for every m 2 N , there exists nm 2 N such that for n  nm there holds 0 u0m  uP ,n

in ‚0n .

(5.3.33)

By (5.3.30) and (5.3.29), u.x/  uP ,n .x/  vP ,1,n .x/ C vP ,2,n .x/ By (5.3.32),

8x 2 ‚n , 8n 2 N .

vP0 ,1,n .x/  u.x/ 8x 2 ‚0n \ .

(5.3.34) (5.3.35)

Moreover vP ,2,n .x/ remains bounded independently of n when x remains in a small enough neighborhood of any compact subset of @ \ QP . If nk is a sequence such that uP ,nk , vP ,1,nk and vP ,2,nk converges locally uniformly in ‚ D QP \ , and uP , vP ,1 and vP ,2 are respectively the corresponding limits, it follows from (5.3.34), (5.3.35) and the fact that vP0 ,1,n .x/ D vP ,1,n .x  2n1 eN /, vP ,1 .x/  u.x/  u.x/  uP .x/  vP ,1 .x/ C vP ,2 .x/

8x 2 ‚.

Thus u is a large solution and the conclusion follows from Theorem 5.3.3.

(5.3.36) 

Remark 5.4. Since the assumptions on the regularity of the boundary do not imply that the maximal solution uN is a large solution, the above proof shows that uN  u is bounded in , where u is obtained in (5.3.26). This follows from the fact that uP ,2,n remains locally bounded on any compact subset of 1 .

182

Chapter 5 Classical theory of maximal and large solutions

Remark 5.5. Under the assumption that @ is locally a continuous graph, it satisfies ı

 D .

(5.3.37) c

If this identity holds, we say that  is outer reachable. Equivalently @ D @ , thus c for any a 2 @, there exists a sequence ¹an º   which converges to a. Furthermore there exists a decreasing sequence of smooth domains ¹0n º such that \n 0n D . Clearly, if  is an outer reachable bounded domain the function u defined by .5.3.26/ is a solution of (5.3.13) in . We conjecture that if the maximal solution u is a large solution then u is a large solution.

5.3.2 Applications to power and exponential types of nonlinearities For q > 1, we consider

u C jujq1 u D 0

(5.3.38)

in   RN . As an applications of the previous results, we have: Theorem 5.3.5. Assume that N  2 and   RN is an outer reachable bounded domain. Then for any 1 < q < N=.N  2/ (q > 1 if N D 2), equation .5.3.38/ admits a unique positive large solution in . c

Proof. If y is @ there exists a sequence ¹yn º   such that yn ! y as n ! 1. By Theorem 5.1.6, equation .5.3.38/ admits a positive maximal solution u. By the Keller–Osserman estimate, any solution u satisfies u.x/  C.N , q/ ..x//2=.q1/

8x 2 .

(5.3.39)

Let v1 be the unique function satisfying in B1 n ¹0º  v C v q D 0 v D0 on @B1 1=.q1/ lim jxj2=.q1/ v.x/ D N ,q

(5.3.40)

jxj!0

where N ,q D 2.2q  N q C N /=.q  1/2 . Then v1 is radially symmetric. Since c yn 2  and u is a positive large solution, v1 ..  yn /  u in  \ B1.yn /. Letting yn ! y, yields v1 .x  y/  u.x/

8y 2 @, 8x 2  \ B1 .y/.

This implies u.x/  c ..x//2=.q1/

8x 2 , s.t. .x/  1=2.

The conclusion follows from Theorem 5.3.2.

(5.3.41) 

183

Section 5.3 Uniqueness of large solutions ı

Remark 5.6. The assumption that  D  is necessary. Actually, if there exists a point y 2  n  such that B .y/ n ¹yº  , there exist infinitely many large solutions u of .5.3.38/. For example, they may differ by the value of the limit lim

x!y

u.x/ D k 2 .0, 1. N .x  y/

If k D 1, then u.x/ D N ,q jx  yj2=.q1/ .1 C ı.1// when x ! y. Remark 5.7. The assumption of positivity of the large solutions in Theorem 5.3.5 is unnecessary if @ is compact. If @ is not compact, and if for any ı > 0 small enough the set ı :D ¹x 2  : .x/ > ıº has a countable number of connected components, some of them with a non-compact boundary, the proof of Theorem 5.3.5 fails. Theorem 5.3.6. Let   RN be a domain with a compact boundary @ which is locally the graph of a continuous function. Then, for any q > 1, equation .5.3.38/ admits at most one large solution. Proof. If  is compact, this is a direct application of Theorem 5.3.3. If  is an exterior domain, any solution decays at most like cjxj2=.q1/ when jxj ! 1. Comparing with any constant ˙ ( > 0), it follows that any large solution is positive in . It is clear from the proof of Theorem 5.3.3 that two large solutions u and u0 , if they exist, must satisfy u.x/ lim D 1. (5.3.42) .x/!0 u0 .x/ Therefore, combining this estimate, with the decay at infinity, we derive that u0 is dominated by the supersolution .1 C /u C . Letting ! 0 yields u0  u. The result follows.  If we apply our general uniqueness results to equation u C e au D 0.

(5.3.43)

with a > 0, we obtain uniqueness of the large solution in the situations of Theorem 5.3.2 and Theorem 5.3.3. ı

Theorem 5.3.7. Assume that   R2 is a bounded domain such that  D . Then equation .5.3.43/ admits a unique large solution in . Proof. By the Keller–Osserman estimate any solution of .5.3.43/ satisfies u.x/ 

3 2 ln .1=.x//  ln a C N a 2

8x 2 .

(5.3.44)

184

Chapter 5 Classical theory of maximal and large solutions c

If  is bounded and d iam./ D R=3, for any yn 2  close enough to @,   BR .yn /. Therefore, any large solution is larger than the restriction to  of vk ..  yn /, where vk is the solution of 1 v 00  v 0 C e av D 0 on .0, R/ r v.R/ D ˇ lim v.r /= ln r D k;

(5.3.45)

r !0

existence is ensured from Lemma 5.2.5, for any k < 2=a, and ˇ is arbitrary. Therefore u.x/  vk ..  y/ and u.x/  c ln .1=.x//

8x 2  s.t. .x/ < min¹R=2, 1=2º.

(5.3.46)

Combining this inequality and .5.3.44/ and using Theorem 5.3.2 implies the uniqueness.  Theorem 5.3.8. Let   RN be a domain with a compact boundary @ which is locally the graph of a continuous function. Then equation .5.3.43/ admits at most one large solution. Proof. It is a direct application of Theorem 5.3.3 with L D 1.



For equations of types .5.2.20/ and .5.2.32/ uniqueness results have been obtained in [101, Theorems 7.1 and 7.2] when  is a ball and h.x/ D .1  jxj/˛ C.x/ where C is a continuous and positive function defined in B 1 and ˛ > 2. The proof relies on the precise asymptotic expansion near @B1 . The proofs extend easily to the case where B1 is replaced by a bounded smooth domain , in which cases precise asymptotic expansions are also valid.

5.4 Equations with a forcing term In this section we consider an equation with a forcing term u C g ı u D f .x/

(5.4.1)

where g is locally integrable and non-decreasing,  is any domain and f is a locally integrable function or even a measure.

5.4.1 Maximal and minimal large solutions Theorem 5.4.1. Let  be a domain in RN and let g 2 C.R/ with g.0/  0 be nondecreasing and g satisfies the positive Keller–Osserman condition. Assume also that equation u C g ı u D 0 (5.4.2)

185

Section 5.4 Equations with a forcing term

admits a subsolution v0 in . Then for any f 2 L1loc ./, f  0 equation .5.4.1/ possesses a maximal solution. Proof. Since v0 is a subsolution of (5.4.2), and g satisfies the positive Keller–Osserman condition, this equation admits a maximal solution. Therefore there exists a smallest solution larger than v0. For simplicity we still denote it by v0 and assume that it is a solution in . Let ¹n º be a smooth exhaustion of  and for every n 2 N and m, k > 0 denote by un,m,k the classical solution of u C g ı u D fk :D min¹f , kº uDm

in n on @n .

(5.4.3)

Let vn,m and wn,k be the solutions of v C g ı v D 0 vDm and

w D fk wD0

in n on @n

(5.4.4)

in n on @n

(5.4.5)

respectively. Then un,m,k  vn,m  0 and hence .un,m,k  vn,m / D fk  g.un,m,k / C g.vn,m /  fk . Since un,m,k  vn,m vanishes on @n , it follows that un,m,k  vn,m  wn,k

8m 2 N.

(5.4.6)

Both m 7! vn,m and m 7! un,m,k are increasing and vn,m  un,m,k . Moreover limm!1 vn,m D vn is the minimal large solution of .5.4.2/ in n . Therefore, by (5.4.6), (5.4.7) vn  un,k D lim un,m,k  vn C wn,k . m!1

Since wn,k is bounded and vn is locally bounded it follows that un,k is locally bounded in n . Thus un,k is a large solution of (5.4.3), for every k > 0. The two mappings k 7! un,k and k 7! wn,k are increasing. Hence, letting k ! 1, we obtain, vn  un D lim un,k  vn C wn , k!1

(5.4.8)

where wn is the solution of w D f wD0

in n on @n .

For every  2 Cc2.n /, Z Z .un,k  C g.un,k // dx D n

(5.4.9)

n

fk  dx.

186

Chapter 5 Classical theory of maximal and large solutions

Since g.un,k / " g.un /, f 2 L1.n / and, by (5.4.8) and (5.4.9), un 2 L1loc .n /, it follows, Z Z .un  C g.un // dx D f  dx, n 2 Cc .n /, 

n

for every  2  0. In addition, un  vn and consequently the negative part of un is bounded. Therefore, if C n D n \ ¹un  0º, we obtain Z g.un / dx < 1, 0 C

n

for every  as above. This implies that g.un / 2 L1loc .n / and un is a large solution of (5.4.1) in n . Clearly ¹unº is monotone decreasing and un  v0 in n for any subsolution v0 of (5.4.2); by assumption such a subsolution exists. Therefore uf :D limn!1 un is well defined and it is a solution of (5.4.1) in . Actually uf is the maximal solution of (5.4.1) in . Indeed, if u is a solution of .5.4.1/ then, u  un in n , so that u  uf .  Remark 5.8. By construction, there holds uf  u :D u0 ,

(5.4.10)

for any f 2 L1loc ./, f  0. Therefore, if u is a large solution, it is also the case for uf . One interesting feature is that, if the blow up of f near @ is too strong, equation (5.4.1) can admit only large solutions in . We give a simple example below. Theorem 5.4.2. Assume that  is a bounded smooth domain in  and g 2 C.R/ with g.0/  0 is non-decreasing, g satisfies the positive Keller–Osserman condition and Z 1 2N g.s/s  N 1 ds < 1. (5.4.11) 1

If f 2

L1loc ./

is non-negative and satisfies Z f .x/.x/dx D 1

(5.4.12)

Br .a/\

for all a 2 @ and r > 0, then any positive solution of (5.4.1) is a large solution. Proof. Let u be a positive solution of .5.4.1/. For each n 2 N we denote by un the solution of u C g ı u D Tn .f / in  (5.4.13) uD0 on @, where Tn .f / D min¹n, f º. Since u  0, u  un and the sequence ¹unº is increasing, with a limit uf smaller than u. Furthermore, for each n, r > 0 and a 2 @, un  vn,r which is the solution of v C g ı v D 1Br .a/\ Tn .f / vD0

in  on @.

(5.4.14)

187

Section 5.4 Equations with a forcing term

If k > 0, we can consider r D r` and n D n` , such that r` ! 0, n` ! 1 and Z Tn` .f /.x/.x/dx D k. (5.4.15) Br` .a/\

Using the stability result of Theorem 2.3.7, un` ! uk,a when ` ! 1, where uk,a D ukıa is the solution of u C g ı u D 0 u D kıa Letting k ! 1 implies

u  u1ıa

in  on @.

8a 2 @.

(5.4.16)

(5.4.17)

Since @ is C 2, there exists R > 0 such that for any a 2 @, the ball BR,a tangent to @ at a is included in . Therefore uk,a is larger than the solution u0k,a of u C g ı u D 0 u D kıa

in BR,a on @BR,a .

(5.4.18)

This property holds also if k D 1. Since, by symmetry, lim u01,a .a  tRn/ D 1, t #0

where n D na is the normal outward unit vector to @BR,a (and also @) at a, we obtain lim u.x/ D 1.

.x/!0



Remark 5.9. When @ is not regular enough (not Lipschitz for example), it is not known under what conditions linking g and @, equation (5.4.1) (even with g ı u D jujq1 u with q > 1) admits only large solutions in . Theorem 5.4.3. Assume  is a domain in RN with a compact boundary, g 2 C.R/ with g.0/  0 is non-decreasing and g satisfies the positive Keller–Osserman condition. If (5.4.2) possesses a solution v0 in . Then, for every non-negative f 2 L1loc ./, (5.4.1) possesses a minimal solution larger than v0. If we assume moreover that v0 2 C./ and @ satisfies the Wiener condition at every boundary point, then (5.4.1) possesses a minimal large solution uf larger than v0 . Proof. Let ¹n º be a smooth exhaustion of . For every n, we consider the set Uf ,n of all the solutions of (5.4.1) in n which are larger than v0 and we denote by vn,f the infimum of Uf ,n . Then vn,f is the solution of u C g ı u D f u D v0

in n on @n .

(5.4.19)

188

Chapter 5 Classical theory of maximal and large solutions

Since the restriction of v nC1,f to n is larger than v0 , v n,f  v nC1,f , thus the sequence ¹v n,f º is non-decreasing. Clearly v n,f  uf (the function defined in Theorem 5.4.1). Therefore there exists vf D limn!1 vn,f . It is a solution in  larger than v0. If u is any solution of (5.4.1) larger than v0 , its restriction to any n is larger than vn,f . Thus u  vf . Next, if @ satisfies the Wiener criterion, we can consider for k  max¹v0 .x/ : x 2 @º the solution un,k (un,k,R if  is an exterior domain) of u C g ı u D 1n f uDk u D v0

in  or  \ BR if c  BR on @ on @BR if c  BR .

(5.4.20)

Then un,k (or un,k,R ) is larger than v0 and smaller than any large solution of (5.4.1) larger than v0 . Letting successively k ! 1 and n ! 1 (or R ! 1, k ! 1 and n ! 1) we obtain that uf is a large solution of (5.4.1) larger than v0 and smaller than any large solution larger than v0 . 

5.4.2 Uniqueness In this section, we shall prove that uniqueness of large solutions of (5.4.1) is a consequence of uniqueness of solutions of (5.4.2). The method presented here is an adaptation of previous results concerning the boundary trace of positive solutions of (5.3.1). Theorem 5.4.4. Let  be a domain in RN with a non-empty and compact boundary. Assume that g is non-decreasing, convex and g satisfies the positive Keller–Osserman condition. (i) Let v0 2 C./ be a solution of (5.4.2). If @ satisfies the Wiener condition and (5.4.2) possesses a unique large solution larger than v0 then, for every nonnegative f 2 L1loc ./, (5.4.1) possesses a unique large solution larger than v0. c

(ii) Let  be a bounded domain such that @ D @ . Assume also that g satisfies the weak singularity assumption if N  3, or afC < 1 if N D 2. If (5.4.2) admits a unique large solution, then for every non-negative f 2 L1loc ./, (5.4.1) possesses a unique large solution. Proof. Assertion (i). First, we assume that  is bounded and we consider a smooth exhaustion ¹n º of . For any k 2 N , k  max¹v0.x/ : x 2 @º, and n 2 N , let uk,1 f be the solution of n

u C g ı u D 1n f uDk

in  on @.

(5.4.21)

We also denote by uk,n,f the solution of u C g ı u D f in n u D kn on @n ,

(5.4.22)

189

Section 5.4 Equations with a forcing term

where kn D max¹k, max¹uk,1 f .x/ : x 2 @n ºº. Then uk,n,f  uk,1 f in n . n n If u is any large solution of (5.4.1) in , it is larger than uk,f . Furthermore, for any n, there exists kn such that kn  max¹u.x/ : x 2 @n º. Therefore uk,n,f  u in n . When k ! 1, uk,0 converges to the minimal large solution u of (5.4.2), while uk,1 f converges to the minimal large solution u1 f of n

n

u C g ı u D 1n f

(5.4.23)

in . which is smaller than any large solution of (5.4.1) in . Finally, when n ! 1, u1 f increases and converges to the minimal large solution uf of (5.4.1) in . On n the other hand, uk,n,f ! un,f when k ! 1 and un,f is a large solution of (5.4.22). Furthermore un,f ! uf when n ! 1, and uf is the maximal large solution of (5.4.1) in . We put Zf :D Zk,n,f D uk,n,f  uk,1

n

f

.

Then Zf  0 and .Zf  Z0 / D g.uk,n,f /  g.uk,1

n

f

/  g.uk,n,0 / C g.uk,0/

(5.4.24)

in n . We can rewrite the right-hand side in the form df .uk,n,f  uk,n,0/  df .uk,1 where df D

g.uk,n,f /  g.uk,n,0 / , uk,n,f  uk,n,0

Since uk,n,f  uk,n,0 , uk,1

n

f

df D

n

f

 uk,0 /

g.uk,1

n

uk,1

 uk,0 , uk,n,f  uk,1

f

n

n

f

/  g.uk,0 /

f

 uk,0

.

, uk,n,0  uk,0 and g is

convex, df  df . Therefore the right-hand side of (5.4.24) is larger than df .Zf  Z0 /. On @n , uk,n,f D uk,n,0 and uk,1 f D uk,0 , thus Zf Z0 vanishes. It follows n by the maximum principle that Zf  Z0 in n . This reads uk,n,f  uk,1

n

f

 uk,n,0  uk,0

in n .

(5.4.25)

Letting successively k ! 1 and n ! 1, and using the above convergence properties yields uf  uf  u0  u0 in , (5.4.26) and u0 and u0 are respectively the maximal and the minimal large solution of (5.4.2) in . Since we have assumed that they coincide, it follows that uf D uf . Next, if  is an exterior domain, we consider again a smooth exhaustion n of , where @n is compact and dH .@n , @/ < n1 (we recall that dH is the Hausdorff distance between compact sets) and for R > 1, we set n,R D n \ BR and

190

Chapter 5 Classical theory of maximal and large solutions

R D  \ BR . For k  max¹v0 .x/ : x 2 @º, let uk,1 u C g ı u D 1n,R f uDk u D v0 As in the first case limn!1 limk!1 uk,1

n,R

f

be the solution of

in R (5.4.27)

on @ on @BR .

n,R

f

D u1

R

f

, which is the minimal so-

lution of u C g ı u D 1R f in R larger than v0 , and limR!1 u1

R

f

D uf which is the minimal solution of

(5.4.1) in  larger than v0. We also denote by uk,n,1

R

f

the solution of

u C g ı u D f in n,R u D kn,R on @n

(5.4.28)

u D kn,R on @BR , where kn D max¹k, max¹u1 max¹v0 , u1

R

f

R

f

: x 2 @n ºº. Since u1

º. When k ! 1, and n ! 1 uk,n,1

R

R

f

f

 v0 in , uk,n,1

R

f



converges to the maximal

solution u1 f of (5.4.1) in R and when R ! 1, u1 f decreases to the maximal R R solution uf of (5.4.1) in . Using the same approximation and convexity argument as in the first case, we obtain uk,n,1

R

f

 uk,1

n,R

f

 uk,n,0  uk,0

in n,R .

(5.4.29)

Letting successively k ! 1, n ! 1 and R ! 1 yields (5.4.26) and implies uniqueness. c Assertion (ii). Since @ D @ , there exists a decreasing sequence of bounded 0 smooth domains ¹n º such that  D \n 0n . Furthermore, we can assume that 1 1 < dH .@0n , @/ < . 2n n For integers n, m, k, let uk,n,1

m

f

be the solution of

u C g ı u D 1m f uDk

in 0n on @0n ,

(5.4.30)

where ¹m º is the smooth exhaustion of  already used in (i). Then un,1 f D m limk!1 uk,n,1 f is a large solution of m

u C g ı u D 1m f

(5.4.31)

191

Section 5.4 Equations with a forcing term

in 0n . If we denote u1 f D limn!1 un,1 f , it is clearly a solution of (5.4.31) m m in . If x 2 , we denote by xn0 a point in @0n such that jx  xn0 j D dist .x, @0n /. Since  is bounded, we can assume that diam./ D R=2, therefore, for any n  nR ,   Ba .R/, for any a 2 @0n . Since  is bounded, we can notice that uk,n,1 f is m bounded from below by a constant ˇ independent of n  nR , k  0 and m. If N  3, we denote by v the solution of (5.2.6) with k D 1 (it exists because of the Keller– Osserman condition). If N D 2, let v denote the solution of (5.2.13) with k D 2=agC. Then, for any n  nR , there exists kn such that ukn ,n,1

m

f

.x/  v.jx  xn0 j/,

and it holds if for k  kn and any m. This implies that u1 f is a large solution of m (5.4.31) in  and uf D limm!1 u1 f is a large solution of (5.4.1) in . Furtherm more, any large solution of (5.4.1) in  is larger than the restriction of uk,n,1 f to m . Consequently, it is larger than uf by letting successively k, n, m ! 1. For ` > n, let uk,n,`,1 f the solution of m

u C g ı u D 1m f

in `

u D kn,m

(5.4.32)

on @`

where kn,m D max¹k, max¹uk,n,1 f .x/ : x 2 @` ºº. Then uk,n,`,1 f is larger m m than the restriction of uk,n,1 f to ` . Clearly, u`,1 f D limk!1 uk,n,`,1 f is m m m a large solution of (5.4.31) in ` and u1 f D lim`!1 u`,1 f is a large solution m m of (5.4.31) in  which dominates any large solution of the same equation in the same domain. Thus uf D limm!1 u1 f is the maximal large solution of (5.4.1) in  m (it is a large solution since it is larger than uf ). For completing the proof we use the same method as in assertion (i) that we start in ` in setting Zf :D Zk,n,`,m D uk,n,`,1

m

f

 uk,n,1

m

f

.

Since Zf  Z0 vanishes on @` , the convexity argument already used yields uk,n,`,1

m

f

 uk,n,1

m

f

 uk,n,`  uk,n

in ` .

Letting successively k, n, ` and m go to 1 implies uniqueness.

(5.4.33) 

Theorem 5.4.5. Let  be a bounded domain such that @ is locally a continuous graph. Assume that g is non-decreasing, convex and g satisfies the positive Keller– Osserman condition. If (5.4.2) possesses a large solution (necessarily unique), then, for every non-negative f 2 L1loc ./, (5.4.1) possesses a unique large solution. Proof. With the notations of Corollary 5.3.4, there exists a decreasing sequence of smooth bounded domains ¹0n º such that  D \n 0n . For each n, m, k we denote by

192 u0k,m,1

Chapter 5 Classical theory of maximal and large solutions

n

f

the solution of u C g ı u D 1n f uDk

Then u0k,m,1

n

f

in 0m on @0m .

(5.4.34)

is increasing with respect to k, m and n and lim

lim u0k,m,0 D u,

m!1 k!1

and u is a large solution (the unique one) of (5.4.2) by Corollary 5.3.4. Then lim

lim

lim u0k,m,1

n!1 m!1 k!1

n

f

D uf

is a large solution of (5.4.1) in  and uf  u. Let uk,n,f be the solution of (5.4.22) with u D kn on @n where kn D max¹k, max¹u0k,m,1 f .x/ : x 2 @n ºº. Then uk,n,f  u0k,m,1

n

n

. As in the proof of Theorem 5.4.4 there holds f uk,n,f  u0k,m,1

n

f

 uk,n,0  u0k,m,0 .

This implies (5.4.26) and the conclusion follows from Corollary 5.3.4.

(5.4.35) 

5.5 Notes and comments The existence of a maximal solution to (5.1.1) in a bounded domain goes back to J. B. Keller [60] and R. Osserman [96]. The construction of such a solution in an unbounded domain, in a model case such as f .u/ D e u can be found in [82]. The weak singularity assumption (5.2.7) was introduced by H. Brezis and Ph. Benilan (see Notes at the end of Chapter 2). The exponential order of growth (4.5.14) was introduced by J. L. Vàzquez in his study [112] of two-dimensional nonlinear Poisson equations in R2 . The fact that a maximal solution in a bounded domain which satisfies the Wiener condition is a large solution has always seemed clear, and the question turned out to be of interest when the boundary of the domain is less regular. The idea of using singular solutions to prove that the maximal solution is a large solution was introduced by M. Marcus and L. Véron in their study [76] of u C jujq1 u D 0,

(5.5.1)

however, they did not mention the fact, clear from their construction, that when 1 < q < N=.N 2/ the maximal solution is actually a large solution, a fact which was first observed by L. Véron in [120]. The systematic use of the weak singularity assumption, or the finiteness of the exponential order of growth when N D 2, is due to M. Marcus and L. Véron [82].

193

Section 5.5 Notes and comments

In 1997, J. S. Dhersin and J. F. LeGall introduced a probabilistic method which gave a necessary and sufficient condition in order for the maximal solution of equation (5.5.1) with q D 2 to blow up at a boundary point. This condition is expressed by a Wiener-type condition involving the C2,2-capacity associated to the Sobolev space W 2,2 . The restriction q D 2 follows from the probabilistic tools used, in particular the super-Brownian motion and the Brownian snake, which is not relevant for other exponents. In 2003, D. Labutin gave a partial extension of Dhersin–LeGall’s result to general exponent q > 1 in proving that, if for any boundary point x there holds Z

1

0

  dr C2,q 0 c \ Br .x/ N 1 D 1, r

(5.5.2)

0

where C2,q 0 stands for the W 2,q capacity relative to some large ball B containing , then the maximal solution in  is a large solution. The full extension of Dhersin– LeGall’s result was obtained by M. Marcus and L. Véron [85] who proved that (5.5.2) is a necessary and sufficient condition for the existence of a solution blowing up at a prescribed boundary point x. Furthermore, they proved that if (5.5.2) holds everywhere on the boundary, the maximal solution is actually -moderate which means that it is the limit of an increasing sequence of positive moderate solutions. Capacitary estimates involving the expression Z 0

1

C2=q,q 0 .@ \ Br .x//

dr r N 1

D 1,

(5.5.3)

where C2=q,q 0 stands for the Bessel capacity in dimension N 1, have been used intensively by M. Marcus and L. Véron [80] for proving that if  is a bounded C 2 domain and K a compact subset of @, the maximal solution uK of (5.5.1) which vanishes on @ n K is indeed -moderate. The question of uniqueness of large solutions was considered by B. Loewner and L. Nirenberg for (5.3.38) with q D .N C2/=.N 2/. Using conformal transformations in a smooth bounded domain, they proved that uniqueness holds. Using the self-similar q,0 transformation Tr , I. Iscoe [58] observed that in a star-shaped bounded domain, there exists at most one large solution to (5.5.1). In 1990, C. Bandle and M. Marcus proved in [8] the uniqueness of the large solution of (5.5.1) in a smooth bounded domain . For equations similar to (5.5.1), extension of this uniqueness result, based on asymptotic behavior of large solutions, was performed in 1992 by C. Bandle and M. Marcus, with more general nonlinearities [9] and L. Véron, with more general second order elliptic operators [117]. Bandle and Marcus [10] continued their studies of large solutions treating more general equations (both the linear and nonlinear parts), including equations on manifolds and estimates for the gradient of large solution. Some of their results are presented in Chapter 6. In [76] M. Marcus and L. Véron introduced the local translation method to prove uniqueness of large solutions of (5.5.1) when @ is

194

Chapter 5 Classical theory of maximal and large solutions

locally a continuous graph. While the previously mentioned works established uniqueness of large solutions by deriving sharp asymptotic estimates for their boundary blow up, the method used in [76] was designed to estimate the limit of the ratio of two large solutions when approaching the boundary. The result of [76] was extended to convex nonlinearities in [82] presented here as Theorem 5.3.3. Observe that this theorem does not assume that g.0/ D 0; therefore the result applies in particular to a nonlinearity such as g.t / D e t . The technique used in the proof of Theorem 5.3.2 was first introduced by M. Marcus and L. Véron [77] in order to prove uniqueness of solutions of generalized boundary value problems for equation (5.5.1) in the boundary subcritical case. In [120] it was applied by L. Véron in order to prove uniqueness of large solutions of (5.5.1) in the interior subcritical case, in bounded domains satisfying only the N c . In [76] this technique was extended to non-decreasing, convex condition @ D @ nonlinearities g. In [83] M. Marcus and L. Véron studied the problem of existence and uniqueness of large solutions for equations such as (5.4.1), with convex nonlinearities and locally integrable forcing terms, in unbounded domains.

Chapter 6

Further results on singularities and large solutions

In this chapter we give further results on positive solutions of semilinear elliptic equations in continuation with the topics that were developed in Chapters 3 and 4. We study in particular uniqueness results for weakly superlinear nonlinearities and anomalous results dealing with equations with fading absorption, symmetry results of large solutions in a ball, and sharp asymptotic expansion of large solutions near the boundary.

6.1 Singularities 6.1.1 Internal singularities Assume that g is a continuous non-decreasing function R 7! R, satisfying g.0/  0 and  is a smooth bounded domain in RN containing 0. We have seen in Chapter 2 that a necessary and sufficient condition in order to solve, for k > 0, u C g ı u D kı0 uD0

in  on @,

is that g satisfies the weak singularities assumption Z 1 g.s 2N /s N 1 ds < 1.

(6.1.1)

(6.1.2)

0

Furthermore the solution u :D uk is unique and satisfies limx!0 jxjN 2 u.x/ D cN k, for some cN > 0, and the mapping k 7! uk is increasing. We denote Z 1 G.s/1=2 ds K.a/ D where G.s/ D

Rs 0

(6.1.3)

(6.1.4)

a

g./d ; K.a/ can be finite or infinite.

Proposition 6.1.1. I Assume that K.a/ < 1 for some a > 0, then uk converges, when k ! 1, to u which is the smallest solution of (6.1.6) satisfying limx!0 jxjN 2 u.x/ D 1. II Assume K.a/ D 1 for any a > 0, then we have the following alternative: (i) either uk converges to u, (ii) or limk!1 uk .x/ D 1 locally uniformly in  n ¹0º.

(6.1.5)

196

Chapter 6 Further results on singularities and large solutions

Proof. Notice that the assertion on the function g implies either that g is always positive or there exists a0  0 such that g.a0 / D 0 and g > 0 on .0, 1/. The assumption K.a/ < 1 is a variant of the Keller–Osserman condition. The first assertion follows from Chapters 3 and 4, since any solution of u C g ı u D 0 uD0

in  n ¹0º on @,

(6.1.6)

is bounded from above, in  n ¹0º by the maximal solution u of (6.1.6). This maximal solution is the limit, when ! 0, of the solutions u of u C g ı u D 0 in  n B limjxj! u.x/ D 1 u D 0 on @.

(6.1.7)

Thus u is a solution of (6.1.6) dominated by u and it is the smallest solution satisfying (6.1.5). For the second assertion, we first consider the case of a ball BR . The function uk is Q D uQ k .s/ D uk .r /. Then radial and we set s D r 2N , u.s/ uQ 00 D

s 2.N 1/=.N 2/ g ı uQ .N  2/2

in .S, 1/

Q D cN k. For k > 1, the uk are bounded from below by u1 . and lims!1 u.s/=s This implies that there exists s1 > S such that uQ k .s/, uQ 0k .s/  0 on Œs1 , 1/, for any k  1. Thus all the uQ k are convex on Œs1 , 1/, and either uQ D limk!1 uQ k is defined on Œs1 , 1/, either uQ is defined on Œs1 , s2 / with s2 < 1 and lims!s2 u.s/ Q D 1, or 0 Q uQ 1. In the second case lims!s2 u .s/ D 1 and there holds 2.N 1/=..N 2//

2s 2 2 uQ0 .s/  uQ0 .s1 /  1 This implies

Z

u.s/ u.s10 /

.N  2/2

.G.u.s// Q  G.u.s Q 1 //.

dt  C.s  s1 / p G.t /

8s 2 Œs10 , s2 /.

for some s10 2 .s1 , s2 / and some C > 0. Thus Z 1 dt K.u.s10 // D  C.s2  s10 / < 1. p 0 G.t / u.s1 / This contradicts the assumption. Hence the alternative II holds in the ball BR . Notice that the same alternative holds in any ball BR0 : this is due to the fact that if u solves u C g ı u D 0

(6.1.8)

197

Section 6.1 Singularities

in BR , T˛ Œu defined by T˛ Œu.x/ D u.˛x/ solves T˛ Œu C ˛ 2 g ı T˛ Œu D 0

(6.1.9)

in BR0 with ˛ D R=R0 . If we assume that ˛ < 1, then T˛ Œu is a supersolution for (6.1.8) in BR0 . If uk, denotes the solution of (6.1.1) in , it follows from (6.1.3) that T˛ Œuk,BR  is larger than uk˛2N ,BR0 . Therefore T˛ ŒuBR   uBR0 .

(6.1.10)

Consequently, if uBR has an isolated singularity at 0, uBR0 inherits the same property. Next we assume that limk!1 uk,BR D 1 in BR n ¹0º. If, for R0 > R and k large enough, uk,BR0 is non-negative on @BR – this is the case if g.0/ D 0 – then for any k 0 > k, uk 0 ,BR0  uk,BR . Letting successively k 0 ! 1 and k ! 1 implies limk!1 uk,BR0 .x/ D 1 for all x 2 BR n ¹0º. By the construction above, it implies limk 0 !1 uk 0 ,BR0 .x/ D 1 for all x 2 BR0 n ¹0º. If uk,BR0 .R/ D m < 0, then uk,BR  m is a subsolution, thus uk,BR0 .R/  uk,BR  m in BR n ¹0º. Since m is a decreasing function of k, it follows again limk!1 uk,BR0 .x/ D 1, in BR n ¹0º and then in BR0 n ¹0º. Notice also that this property is invariant by translation: this means that if limk!1 uk,BR .x/ D 1 in BR n ¹0º, then for any R0 > 0 and a 2 RN , limk!1 uk,BR0 .x  a/ D 1 in BR0 .a/ n ¹aº. Finally, we assume that  is a general bounded domain and let R0 > R > 0 such that BR    BR0 . If g.0/ D 0, or more generally if uk,BR , uk,BR0 and uk, remains non-negative, then uk,BR  uk,

in BR and uk,  uk,BR0

in .

(6.1.11)

Thus, if limk!1 uk,BR0 has an isolated singularity at 0, limk!1 uk, has the same property. If limk!1 uk,BR D 1, then limk!1 uk, D 1 locally uniformly in BR n ¹0º. If a 2  \ BR and Ra > 0 is such that BRa .a/ b , then for any > 0 and ` > 0 there exists k,` such that k > k,` H) uk, .x  a/  U,`,a .x/

in BRa .a/

where U D U,`,a is the solution of U C g ı U D `1B .a/ in BRa .a/ U D0 on @BRa .a/. If we let successively k ! 1, then ` ! 1 and ! 0 in order to have ` N !N D n for a fixed n > 0, we infer limk!1 uk, .x  a/  un,BRa .a/ .x/. Since un,BRa .a/ .x/ D un,BRa .x  a/, we let n ! 1 and obtain lim uk, D 1

k!1

in BRa .

198

Chapter 6 Further results on singularities and large solutions

Since any y 2  with .y/ < c can be chained to BR by a finite number of open balls BRai .ai / (0  i  ) such that BRai .ai / b , BRai .ai / \ BRai1 .ai 1 / ¤ ;, BRa0 .a0 / D BRa .a/ and BRa .a / D Bc .y/, we conclude that lim uk, D 1

k!1

in  n ¹0º.

Furthermore, this convergence is uniform on any compact subset of  n ¹0º because any compact set can be covered by a finite chain of balls with the above properties. Finally, if the uk,BR , uk,BR0 and uk, do not remain positive, we replace (6.1.11) by uk,BR  m,k  uk,

in BR and uk,  mR0 ,k  uk,BR0

in ,

(6.1.12)

where m,k is the infimum of uk, on @BR and mR0 ,k the infimum of uk,BR0 on @. The remainder of the proof is easily adapted.  Remark 6.1. When g ı u D jujq1 u with 1 < q < N=.N  2/ or g ı u D e au and a > 0, there holds u D u (see Ch. 3), and the proofs of equality involve sharp expansion estimates and convexity of g. However, it is an open problem whether equality always holds if g is merely convex and satisfies (6.1.2) and K.a/ < 1. We extend this result to another class of nonlinearities. Theorem 6.1.2. Assume g ı u D u.ln C u/˛ with ˛ > 2. Then u D u. The proof is based upon the following proposition. Proposition 6.1.3. Let u be a radial solution of (6.1.6) in BR n ¹0º such that (6.1.5) holds. Then ˛  .N  1/.˛  2/ 2=.˛2/ r C O.r 4=.˛2/ / as r ! 0, r 2=.˛2/ ln u.r / D .˛/ C 2˛ (6.1.13) and r ˛=.˛2/ .ln u.r //r D . .˛//˛=2 C O.r 4=.˛2/ / where



.˛/ D

2 ˛2

as r ! 0,

(6.1.14)

2=.˛2/ .

(6.1.15)

Lemma 6.1.4. Assume that ˛ > 0 and v is a positive solution of N 1 0 (6.1.16) v C v 0 2  v ˛ D 0 in .0, R, r such that limr !0 v.r / D 1. Then for any > 0 there exists r . / 2 .0, R/ such that v 00 C



N 1 v0  1   1 C r v ˛=2 v ˛=2

8r 2 .0, r . /.

(6.1.17)

199

Section 6.1 Singularities

Proof. Since v.r / ! 0 when r ! 1 and v is singular at 0, it follows from (6.1.16) that there exists r0 2 .0, R such that v 0 < 0 on .0, r0. Therefore v 00 C v 0 2  v ˛  0 in .0, r0.

(6.1.18)

We can take v D  as a new variable, set h./ D v 0 2 and obtain 1 0 h C h  ˛ 2

in Œ0 , 1/.

This implies .e 2 h/0  2e   ˛ and hence, for any 2 .0, 1/, there exists . / > 0 such that h./  1  in Œ. /, 1/. ˛ Returning to v it means that vr  1 C v ˛=2

8r 2 .0, r . /.

(6.1.19)

This implies in particular limr !0 v 0 .r / D 1. We set !.r / D v 0 .r /, then ! 00 C

N 1 0 N 1 ! C 2!! 0  !  ˛v ˛1! D 0. r r2

Since ! < 0 on .0, r0, (6.1.20) implies   N 1 00 C 2! ! 0 < 0 in .0, r0. ! C r

(6.1.20)

(6.1.21)

Because !.r / ! 1, ! 0 is positive and thus N 1 0 v r

C v 0 2  v ˛  0 in .0, r0.

Solving this inequality in v 0 yields r  N2r1 

N 1 2r

2

C v ˛  v 0  0.

(6.1.22)

(6.1.23) 

This implies the left-hand side of inequality (6.1.17).

Lemma 6.1.5. Let ˛ > 1 and u be a positive radial solution of (6.1.6) in BR n ¹0º. Then limr !0 r N 2u.r / D 1 if and only if limr !0 r 2=˛ ln u.r / D 1. Q D u.r /. Then Proof. We consider again the change of variable s D r 2N , u.s/ uQ 00 D

1 s 2.N 1/=.N 2/ u.ln Q C u/ Q ˛ .N 2/2

in .S, 1/.

(6.1.24)

If limr !0 r N 2 u.r / D 1, there holds also Q D lims!1 uQ 0 .s/ D 1. lims!1 s 1 u.s/

(6.1.25)

200

Chapter 6 Further results on singularities and large solutions

Since uQ is convex, (6.1.25) implies u.s/ Q  s uQ 0 .s/.1 C o.1// and  ˛ .ln u/ Q ˛  ln s C ln uQ 0 .s/ C O.1/  .N  2/2 .ln s/˛ .ln uQ 0 /˛ for s large enough. Plugging these estimates into (6.1.24) yields uQ 00  2s N=.N 2/ .ln s/˛ .ln uQ 0 /˛ .

(6.1.26)

Moreover, as ˛ > 1, Z

1 s

Z

and

1 s

uQ 00 1 .ln uQ 0 .s//1˛ , dt D 0 0 ˛ uQ .ln uQ / ˛1

t N=.N 2/ .ln t /˛ dt  As 2=.N 2/ .ln s/˛ ,

for some A > 0 and s large enough. This implies ln uQ 0 .s/  Bs 2=..N 2/.˛1// .ln s/˛=.1˛/ for some B > 0. Thus, for any > 0 there holds for s large enough, u.s/ Q  es

.2/=.N 2/.˛1/

,

and finally ln u.r /  r .2/=.˛1/ ,

(6.1.27)

for r small enough. It is inferred from this that limr !0 r 2=˛ ln u.r / D 1. Clearly the reverse implication is true.  Proof of Proposition 6.1.3. By Lemma 6.1.4 and Lemma 6.1.5, we have v 0 .r / D 1, r !0 v ˛=2 .r / lim

where v D ln u. This implies  lim r

2=.˛2/

r !0

and

v.r / D

2 ˛2

2=.˛2/ D .˛/

lim r ˛=.˛2/v 0 .r / D . .˛//˛=2 .

r !0

(6.1.28)

(6.1.29)

Furthermore .N  1/.˛  2/ ˛1 N 1 0 v .r / D  v .r /.1 C o.1//. r 2

(6.1.30)

201

Section 6.1 Singularities

Plugging this estimate into (6.1.16) implies v 00 C v 02 D v ˛ C C v ˛1 .1 C o.1//,

(6.1.31)

where C D .N  1/.˛  2/=2. Taking again  D v.r / as the variable and h./ D v 02 , we see that h satisfies  ˛  1 2 0 2 (6.1.32)  C C ˛1.1 C o.1// , 2 .e h.// D e and thus

  h./ ˛ 1 .1 C o.1// D1C C  ˛ 2 

Writing AD then (6.1.33) becomes

as  ! 1.

(6.1.33)

˛  .N  1/.˛  2/ ˛ C  D , 4 2 4 v0 v ˛=2

D 1 C

A .1 C o.1//, v

which in turn implies v.r / D .˛/r 2=.2˛/ .1 C o.1// and leads to v 0 .r / Ar 2=.˛2/ .1 C o.1//. D 1 C

.˛/ v ˛=2 .r /

(6.1.34)

By integration on .0, r / for r small enough, we obtain v.r /  .˛/r 2=.2˛/ D    Next, since v 0 D v ˛=2 1 C O v1 , we have

2A .1 C o.1//. ˛

(6.1.35)

   1 N 1 0 ˛1 .r / 1 C O v .r / D C v . r v

This estimate improves (6.1.31) since it becomes v 00 C v 02 D v ˛ C C v ˛1 C O.v ˛2 /. Returning to the variable  and h./ D v 02 , we obtain   h./ 2C  ˛ 1 , D 1 C C O ˛  2 2 and

  1 v 0 .r / A D 1 C C O 2 . v  v ˛=2 .r /

(6.1.36)

(6.1.37)

(6.1.38)

202

Chapter 6 Further results on singularities and large solutions

Using the fact that v.r / D .˛/r 2=.2˛/ .1 C O.r 2=.˛2/ //, we derive   v 0 .r / Ar 2=.˛2/ C O r 4=.˛2/ . D 1 C ˛=2

.˛/ v .r / By integration v.r / D .˛/r 2=.2˛/ C

(6.1.39)

  2A C O r 2=.˛2/ . ˛ 

This implies (6.1.13).

Remark 6.2. The two previous lemmas and Proposition 6.1.3 are valid if N D 2. In (6.1.5) and the statement of Lemma 6.1.5, jxj2N has to be replaced by ln.1=jxj/. In the proof of Lemma 6.1.5 the change of variable s D r 2N has to be replaced by r D e t . Proof of Theorem 6.1.2. Let R > 0 such that B R  . We denote respectively by uR and uR the smallest and the largest solution of u C u.lnC u/˛ D 0

(6.1.40)

in BR n ¹0º vanishing on @BR and satisfying (6.1.5). Set M D ¹max u.x/ : jxj D Rº. Since uR is the limit of the solutions uR, of (6.1.7) in BR nB , there holds uR  uR, in BR n B . By letting ! 0, this implies uR  u  u  uR C M

in BR n ¹0º.

(6.1.41)

Since both uR and uR satisfy (6.1.5), they satisfy (6.1.13), from which equality it follows   / / ln uuR .r D O.r 2=.˛2// H) limr !0 uuR .r D 1. (6.1.42) .r / .r / R

R

u/˛

is convex, for any > 0 .1 C /uR is a supersolution which is Since u 7! u.ln C larger than uR near 0. Thus .1C /uR  uR , and finally uR D uR . This implies again limr !0

u.r / u.r /

D 1.

(6.1.43)

Then the equality u D u follows by the same standard convexity argument.



Corollary 6.1.6. Let uk be the solution of (6.1.1) with g ı u D u.lnC u/˛ . Then (i) if ˛ > 2, limk!1 uk D u (ii) if 0 < ˛  2, limk!1 uk D 1. Proof. The first statement is a consequence of Theorem 6.1.2. Next we assume 0 < ˛  2, and then K.a/ D 1 for any a > 0, and that statement (ii) does not hold. Then, by Proposition 6.1.1, u1 D limk!1 uk is a solution of (6.1.6) with g ı u D

203

Section 6.1 Singularities

u.lnC u/˛ , and it satisfies limx!0 jxjN 2 u1 .x/ D 1. Let R > 0 such that B R  ; we set M k D max¹uk .x/ : jxj D Rº and mk D min¹uk .x/ : jxj D Rº. There holds k uk,R  mk  uk  uk,R C MC ,

where uk,R,˛ D u satisfies u C u.ln C u/˛ D kı0 uD0

in BR on @BR .

(6.1.44)

1 u1,R  m1   u1  u1,R C MC .

(6.1.45)

Letting k ! 1 we obtain

First we consider the case ˛ > 1. By Lemma 6.1.5 and (6.1.45), lim r 2=˛ ln.u1,R .r // D 1.

r !0

We set v D lnC u1,R , then limr !0 r v ˛=2.r / D 1. Thus, for any > 0, there exists r 2 .0, R/, such that 1  

v0 v ˛=2

 1 C

on .0, r .

This implies .1  /.r  r / 

 2  1˛=2 v .r /  v 1˛=2 .r /  .1 C /.r  r /, 2˛

which contradicts the fact that limr !0 v.r / D 1. Finally, if 0 < ˛  1, then u.lnC u/˛  u.lnC u/2 C 1. Therefore the solution uk,R,˛ of (6.1.44) is larger than the solution U of u C u.lnC u/2 C 1 D kı0 uD0 which in turn is larger than uk,R,2  

, where D1 D0

in BR on @BR ,

satisfies

in BR in @BR .

Since limk!1 uk,R,2 D 1, we conclude with (6.1.45).



As a consequence of (6.1.13) and Theorem 5.3.2, we have the following result dealing with uniqueness of large solutions. ı

Corollary 6.1.7. Let   RN be a bounded domain such that  D , then equation (6.1.40) admits no large solution if 0 < ˛  2, and a unique one if ˛ > 2.

204

Chapter 6 Further results on singularities and large solutions

Proof. In the case ˛ > 2 the proof is based upon the following estimate which is satisfied by any solution in : 2=.2˛/

u.x/  C˛,R e .˛/..x//

8x 2  such that 0 < .x/  R.

(6.1.46)

This estimate is a consequence of Theorem 4.1.2, but it is easier to use Theorem 6.1.2 and Proposition 6.1.3 and to construct one-dimensional barriers. If u is a solution in a ball BR and if UR satisfies U 00 C U.lnC U /˛ D 0 in .0, R/ U 0 .0/ D 0 limr !R U.r / D 1,

(6.1.47)

there holds u.x/  UR .jxj/ Furthermore

Z Rr D

1 UR .r /

where

Z

s

G˛ .s/ D 2

8x 2 BR .

(6.1.48)

ds p 2G˛ .s/

(6.1.49)

t .ln t /˛ dt .

1

But G˛ .s/ D s 2 .ln s/˛  ˛G˛1 .s/  s 2 .ln s/˛ , thus lnC u.x/  lnC UR .jxj/  .˛/.R  jxj/2=.˛2/ .

(6.1.50)

Consequently, if u is a solution of (6.1.40) in , it satisfies lnC u.x/  .˛/ ..x//2=.˛2/

8x 2 .

ı

(6.1.51) c

Because  D , any boundary point a is the limit of a sequence ¹an º   converging to a. Therefore any large solution u is bounded from below by u1 ..  a/ where u1,BR D limk!1 uk,BR , uk,BR satisfies (6.1.44) and R > diam . In particular, if x 2  and a 2 @ is a projection of x, we obtain by (6.1.13) lnC u.x/  lnC u1,BR ..x//  .˛/ ..x//2=.˛2/ C O.1/.

(6.1.52)

Consequently the inequality (5.3.8) of Theorem 5.3.2 is fulfilled and this implies uniqueness. 

205

Section 6.1 Singularities

6.1.2 Boundary singularities Let g be a continuous non-decreasing function R ! R such that g.0/  0 and  a C 2 bounded domain whose boundary contains 0. We have seen in Chapter 2 that a sufficient condition for solving u C g ı u D 0 u D kı0

in  in @,

is the weak boundary singularity assumption Z 1 g.s 1N /s N ds < 1.

(6.1.53)

(6.1.54)

0

The solution uk is unique and bounded from below by m :D inf¹u0 .x/ : x 2 º (u0 is the solution of (6.1.53) with k D 0). If we denote by † D ¹x 2  : .x/ D º ( > 0), then Z lim uk .x/dS D k 8r > 0, (6.1.55) !1 † \B  r

and the mapping k 7! uk is increasing. Furthermore, if ¹fn,k º is a sequence of non-negative functions in L1.; / with support in  \ Bn , where n ! 0 and Z fn,k dx D k 8n 2 N, 

then the sequence of solutions ¹un,k º of un,k C g ı un,k D fn,k un,k D 0

in 2  on 2 @,

(6.1.56)

converges to uk in L1./ and locally uniformly in , and g ı un,k ! g.uk / in L1.; /. This stability statement is proved in Chapter 2. It leads to the following comparison result. Lemma 6.1.8. Let   0 be two open bounded C 2 domains such that 0 2 @\@0 . Assume that g is non-decreasing, satisfies (6.1.54) and g.0/  0. If uk, and uk,0 are the solutions of (6.1.53) with  and 0 respectively, there holds uk,  uk,0 in . Proof. Let ¹ nº be a sequence decreasing to 0 and fn,k defined by

1 where N D

R

B1C

fn,k D k N nN 1 1\Bn yN dx and B1C D ¹y D .y 0 , yN / 2 B1 : yN > 0º. Then Z 

fn,k dx D k.1 C o.1//.

206

Chapter 6 Further results on singularities and large solutions

Put fQn,k the extension of fn,k by zero in 0 n  and let un,k, and un,k,0 be the solutions of (6.1.56) with  and 0 respectively (in which case fQn,k replaces fn,k ). Since un,k,0 is non-negative, it follows that un,k,  un,k,0 in . Because @ and @0 are C 2 and tangent at 0, there holds .x/  @0 .x/  .x/ C O. n2 / Therefore Z Z fn,k dx  

0

fQn,k @0 dx D

8x 2  \ Bn .

Z

Z 

fn,k @0 dx 



fn,k dx C O. n /.

These inequalities imply that un,k, ! uk, and un,k,0 ! uk,0 when n ! 1 and the claim follows.  Lemma 6.1.9. Assume g 2 C.R/ is continuous, non-decreasing and satisfies g.0/  0 and (6.1.54). Then for any k > 0, R > 0 and m > 0, there exists a unique function v D vk,R,m 2 C 2 ..0, R/ satisfying v 00 

N 1 0 N 1 v C v C g ı v D 0 in .0, R/ r r2 v.R/ D m lim r

N 1

r !0

(6.1.57)

v.r / D k.

Proof. We first notice that the linear ODE N 1 0 N 1 Y 00  Y D0 Y C r r2 admits the two linearly independent solutions Y1 .r / D r 1N and Y2 .r / D r . Assume that v and vQ are two solutions of (6.1.57), then w D v  vQ satisfies N 1 0 N 1 w C w C d.r /w D 0 r r2 where d is non-negative. Since w.r / D o.Y1 .r // as r ! 0 and w.R/ D 0, we conclude that w 0 by the maximum principle. For existence we set w 00 

sD

v.r / D r 1N .s/, This transforms problem (6.1.57) into   1N 1N  00 C cN s N g cN s N D 0

in .0, S/ :D .0, RN =N /

.S/ D RN 1 m

.0/ D k, where cN D N .1N /=N .

rN . N

(6.1.58)

207

Section 6.1 Singularities

For > 0 we denote by  the solution of   1N 1N  00 C cN .s C / N g cN .s C / N D 0 in .0, S/ :D .0, RN =N /

.S/ D RN 1 m

(6.1.59)

.0/ D k. Since g is non-negative,  is convex, hence min¹RN 1 m, kº   .s/  max¹RN 1 m, kº. As in the proof of Lemma 5.2.5, we deduce by a double integration that, for any 0 < s < s 0 < S, there holds Z s0 Z S   ˇ ˇ 1N 1N ˇ  .s/   .s 0 /ˇ  C.s 0 s/CcN . C / N g cN . C / N  ./ d dt . t

s

N 1

Setting c D cN max¹R m, kº and Z SC1 Z SC1  1N  1N ‰c .s/ D  N g c N d dt s

we obtain

t

Z

SC1

lim ‰c .s/ D

s!0



1 N

 g c

1N N



Z

S0

d D N

0

8s 2 .0, S C 1,    N g c 1N d .

0

This right-hand side of the above inequality is finite by assumption, therefore the family of functions ¹  º is equicontinuous on Œ0, ı and there exists a subsequence n with n ! 0 and 2 C.Œ0, S/ as a solution of (6.1.58), such that n ! uniformly on Œ0, S.  Lemma 6.1.10. Assume that the assumptions of Lemma 6.1.9 are verified and K.a/ D 1 for any a > 0. Then the following alternatives hold: (i) either limk!1 vk,R,m D 1, locally uniformly in BR ; (ii) or limk!1 vk,R,m D v R,m , and vR,m is a solution of v 00 

N 1 0 N 1 v C g ı v D 0 in .0, R/ v C r r2 v.R/ D m lim r

r !0

N 1

(6.1.60)

v.r / D 1.

Proof. The proof is analogous to the one of Proposition 6.1.1 in the radial case, since K.a/ D 1 implies that any solution of the Cauchy problem   1N 1N  00 C cN s N g cN s N D 0 in .˛, S/

.S/ D m1  0, 0 .S/ D m2  0 defined on an interval .˛, S/  Œ0, S can be extended up to s D 0.



208

Chapter 6 Further results on singularities and large solutions

Remark 6.3. As in the proof of Proposition 6.1.1 the above alternative does not depend on R. Finally, we have the following analog of Proposition 6.1.1 where K.a/ is defined in (6.1.4). Proposition 6.1.11. Assume that K.a/ < 1 for some a > 0; then uk converges, when k ! 1, to u which is the smallest solution of u C g ı u D 0 in  vanishing on @ n ¹0º, satisfying Z lim u.x/dS D 1 !1 † \B  r

(6.1.61)

8r > 0.

(6.1.62)

Proof. Clearly u exists and it is a locally bounded solution of (6.1.61). Since u  m, it follows from Theorem 3.1.8, Theorem 3.1.12 and Definition 3.1.13 that u admits a boundary trace which is a positive Borel measure  and (6.1.62) holds. For any ı > 0, uk , and therefore u, is dominated in  n Bı by the solution u D uı ,R of  u C g ı u D 0 uD0 lim u.x/ D 1.

in  n Bı on @ n Bı ,

(6.1.63)

x 2  n Bı jxj ! 

This implies that the boundary trace of u has the point ¹0º for singular part and the zero measure on @n¹0º for regular part. Let u be any solution of (6.1.61) in  which vanishes on @ n ¹0º and satisfies (6.1.62). Then for any k > 0 and < r , u is larger in  :D ¹x 2  : .x/ > º than the solution v,r of  v C g ı v D 0 in  v D u1† \Br on † .

(6.1.64)

Since (6.1.62) holds, for any k > 0 and > 0 small enough there exists r . / 2 .0, r  such that Z u.x/dS D k. † \Br./

Then u  v,r ./ in  . Moreover lim!0 v,r ./ D uk by Theorem 2.3.6. Since k is arbitrary it implies u  u.  When K.a/ D 1, only partial results are available. Proposition 6.1.12. Assume  D BR \ H where H is a half-space and 0 2 @H , g is C 1 , convex and non-decreasing and K.a/ D 1 for any a > 01 . Then the following alternative holds: 1

The assumption that g is C 1 is unnecessary but it simplifies the exposition of the proof since it allows to differentiate the equation instead of working with difference quotients.

209

Section 6.1 Singularities

(i) either uk converges to u, which is a solution in  or in  n ¹Lº where L is a segment normal to @ at 0, (ii) or limk!1 uk .x/ D 1 locally uniformly in . Proof. By dilation and rotation, we can assume that R D 1 and H :D ¹y D .y 0 , yN / : yN > 0º. Step 1: Geometric properties of the uk . Because the uk are unique, they depend only on r D jyj and the latitude angular variable D sin1 .yN =jyj/ 2 Œ0, =2. By Theorem 2.3.6, uk is the limit, when j ! 1, of the sequence ¹uk,j ºj 1 of solutions of u C g ı u D fk,j in B1C uD0 on @B1C

(6.1.65)

where fk,j is non-negative, has compact support in BjC1 and satisfies Z B C1

0 fk,j .y/yN dy D cN k,

j

0 > 0. We can also assume that fk,j depends only on r and and for some constant cN is non-decreasing in . In spherical coordinates equation (6.1.65) takes the following form

ur r 

N 1 ur r



1  u r2 

C g ı u D fk,j uD0

N 1 in .0, 1/ SC N 1 in .0, 1/ @SC .

(6.1.66)

Since u D u.r , /,  u D u C .N  2/ tan u . If we set U D u , it satisfies    N 2 N 1 1  0 Ur r  Ur  2 U C .N  2/ tan U C C g .u/ U D fk,j  . r r sin2

Since hk,j  0 and U D u is non-negative on @B1C and satisfies U C d U D hk,j :D fk,j  where d.x/ D g 0 .u/ C

in B1C

N 2  0, cos2

it follows by the maximum principle that U  0 in B1C . Because uk,j ! uk locally C

uniformly in B 1 n ¹0º, uk inherits the same property, and 7! uk .r , / is maximal for D 0. Clearly u D limk!1 uk shares this property, even if it achieves infinite values. Differentiating the equation satisfied by u with respect to yj (1  j  N  1), we obtain uyj C g 0 .u/uyj D fk,j y in B1C . j

210

Chapter 6 Further results on singularities and large solutions

Since u vanishes on @B1C and is positive in B1C , uyj  0 on @B1C . Using the fact that fk,j y  0, we derive from the maximum principle that uyj  0 in B1C . Since j is j arbitrary and the problem is invariant with respect to rotations in the hyperplane H , it follows that r 7! uk .r , / is decreasing, and so is u. Finally, we prove that uk is radially decreasing. In order to approximate the uk , we can replace the uk,j by the uk, ( > 0) where uk, solves u C g ı u D 0 in B1C n BC u. , ./ D kPRN on @BC uD0

C

on @B1C [ @H \ B1

where PRN D yN jyjN is the Poisson kernel in RN . We set C , and clearly uk  PRN C C t D ln r and uQ k, .t , / D uk, .r , /, then uQ k, satisfies uQ t t C .N  2/uQ t C  uQ D e 2t g ı uQ u.ln Q , ./ D ˇN k 1 , u.0, Q ./ D 0 u.t Q , ./ D 0

N 1 in .ln , 0/ SC N 1 on SC N 1 on Œln , 0/ @SC .

Q ./  0 and w.ln Q , ./  0 and Set wQ D uQ t , then w.0, Q wQ D 2e 2t g ı u. Q wQ t t  .N  2/wQ t   wQ C e 2t g 0 .u/ Since g 0  0, the linear operator on the left-hand side satisfies the maximum principle and since 2e 2t g.v/ Q  0, it follows that wQ  0. This implies the claim. Step 2: Case where u is finite. We write the equation satisfied by uk in spherical coordinates .r , / 2 RC S N 1 ur r 

N 1 r ur



1  u r2 

CgıuD0 uD0

N 1 in .0, 1/ SC N 1 . on .0, 1/ @SC

1,2 N 1 eigenvalue If 1 is the first eigenfunction of R  in W0 .SC / with corresponding R 1 D N  1, normalized by S N 1 1dS D 1, then zk .r / D S N 1 uk .r , ./ 1dS C C satisfies N 1 0 N 1 z C g ı z  0 in .0, 1/ z C z 00  r r2 since g is convex, and limr !0 r N 1 zk .r / D k. By the maximum principle zk is dominated in .0, 1/ by the solution Zk of

Z 00 

N 1 0 r Z

C

N 1 Z r2

C g ı Z D 0 in .0, 1/ Z.1/ D 0 N 1 Z.r / D k; lim r

(6.1.67)

r !0

such a solution Zk exists by Lemma 6.1.9. By Lemma 6.1.10 we can assume that we are in the case where Z :D limk!1 Zk remains locally bounded in .0, 1, and the

211

Section 6.1 Singularities

same property is verified by z :D limk!1 zk . This implies that for any ı 2 .0, 1/, there exists Mı > 0 such that Z u.r , ./ 1 dS  Mı 8r 2 Œı, 1. (6.1.68) N 1 SC

The function 1 depends only on and is increasing on Œ0, =2. Therefore the previous inequality implies that for any  2 .0, =2/, there holds u.r , /  Mı ,

8.r , / 2 Œı, 1 Œ, =2.

Combined with (6.1.68) it implies that there exists Cı > 0 such that Z u.r , ./dS  Cı 8r 2 Œı, 1. N 1 SC

(6.1.69)

By the maximum principle u is smaller than the solution Xı of X D 0 X D0 X D u.ı, ./

in B1C n BıC   C C on @B1C [ @RN C \ B1 n Bı on @BıC .

By the estimates on the Poisson kernel, max¹Xı .x/ : jxj D r º 

C .jxj  ı/N 1

Z N 1 SC

u.ı, ./dS

8ı  jxj  1.

Combining this estimate with (6.1.69) we infer that uk remains locally bounded in B1C n ¹0º, uniformly with respect to k. Thus uk ! u which is a locally bounded solution. Step 3: Case where u is not finite. We suppose limk!1 uk D u is not everywhere finite, neither everywhere infinite in B1 . Either u D 1 only on a segment of the axis eN , with 0 <  < 0, or there exists some x0 D .r0, 0 / 2 B1C with 0 2 .0, =2/ where u.x0 / D 1, then u.x/ D 1 on the conical domain 1 :D ¹x D .r , / : 0 < r  r0 , 0   =2º. Consider now two balls B˛ .z/ and Bˇ .z/ with 0 < ˛ < ˇ and such that B˛ .z/  1 and Bˇ .z/  B1C . Then for any ` > 0, there exists some k > 0 such that uk  ` on @B˛ .z/. Therefore uk is larger in Bˇ .z/ n B˛ .z/ than the function uz,˛,ˇ ,` .x  z/ where u D uz,˛,ˇ ,` satisfies u00  N t1 u0 C g ı u D 0 in .˛, ˇ/ u.ˇ/ D 0 , u.˛/ D `. Since K.a/ D 1, any maximal solution uc of the above ODE, with uc .ˇ/ D 0 and u0c .ˇ/ D c 2 R is defined on whole .0, ˇ. Therefore, lim`!1 uz,˛,ˇ ,` D 1.

212

Chapter 6 Further results on singularities and large solutions

This implies that u D 1 in Bˇ .z/. Since ˛ can be taken arbitrarily small, we can take any z 2 1 , and any ˇ < dist .z, @B1C / :D min¹1  jzj, hz, eN iº. We set r1 D r0 Cmin¹1r0 , r0º and denote by 2 the intersection of BrC1 with the cone of revolution with vertex 0, axis eN and containing the balls Bˇ .z/ with z 2 1 and ˇ < min¹1  jzj, hz, eN iº. Replacing 1 by 2 we see that u D 1 in any ball Bˇ .z/ where z 2 2 and ˇ < min¹1  jzj, hz, eN iº. Iterating this process, we construct an increasing sequence of conical open domains n  B1C with vertex 0 in which u D 1 such that [n n D B1C . This implies that u D 1 in B1C . This concludes the proof.  Remark 6.4. It appears clear that the blow-up segment L is relatively closed and therefore coincides with the whole line ¹eN : 0    1. In the particular case where g ı u D u.ln uC /˛ , we have the following analog of Corollary 6.1.6: Theorem 6.1.13. Let ˛ > 0 and  be a bounded C 2 domain whose boundary contains 0. Then for any k > 0 there exists a unique solution uk to problem in  u C u.lnC u/˛ D 0 u D kı0 on @.

(6.1.70)

If ˛ > 2, limk!1 uk D u, and u 2 C. n ¹0º/ is the smallest solution vanishing on @ n ¹0º satisfying (6.1.62). If 0 < ˛  2, limk!1 uk D 1. Proof. The existence of uk is guaranteed by the fact that (6.1.54) holds. If ˛ > 2, K.a/ < 1, thus the first statement is a consequence of Proposition 6.1.11. For the second statement we assume first that  D B1C . If PB C is the Poisson kernel in B1C , then 1

PB C .x, 0/ D cN 1 ./.r 1N  r /, 1

where x D .r , /. By the maximum principle uk .x/  kPB C .x, 0/  kcN .jxj1N  jxj/ :D kP .x/. 1

Therefore uk is a supersolution for the problem in B1C v C ln˛ .kP ../ C 1/ v D 0 v D kı0 on @B1C ,

(6.1.71)

and the solution vk of (6.1.71), which is dominated by uk , satisfies vk .x/ D kcN r 1N 1./.1 C o.1// as x ! 0. Let ¹˛k º  RC be a sequence which will be made more precise later on, and wk .y/ D wk .˛k x/ D vk .x/. If we set Pk .y/ D cN .jyj1N  ˛kN jyj/ D cN . 1N  ˛kN /, then wk satisfies   in B˛Ck w C ˛k2 ln˛ k˛kN 1 Pk ../ C 1 w D 0 (6.1.72) N 1 w D k˛k ı0 on @B˛Ck .

213

Section 6.1 Singularities

Vk Vk C C .y, 0/ where P C .y, z/ is the Poisson potential in B˛k B˛k B˛k   Vk in B˛Ck with Vk .y/ D ˛k2 ln˛ k˛kN 1 Pk .y/ C 1 . Then

Thus wk .y/ D k˛kN 1 P for the operator  C  Vk .y/ D

ln.k˛kN 1 Pk .y/ C 1/

˛

2=˛

 D

˛k ln.k˛kN 1 / 2=˛

˛k

2=˛

We choose ˛k such that ˛k 

ln.Pk .y/ C k 1 ˛k1N /

C

2=˛

D ln.k˛kN 1 / and we obtain ˛

2=˛

˛k

where bk ./ D

.

˛k

ln.Pk .y/ C k 1 ˛k1N /

Vk .y/ D 1 C

(6.1.73)

˛

D .1 C bk .jyj//˛ .

  ln cN . 1N  ˛kN / C k 1 ˛k1N 2=˛

(6.1.74)

.

(6.1.75)

˛k

A standard asymptotic expansion shows that   2 ˛k D ln˛=2 k 1 C ˛ .N4 2/ lnlnlnkk .1 C o.1//

as k ! 1.

(6.1.76)

k .., 0/ is increasing and it converges, at least formally, to P 1N .., 0/. When k ! 1, P VC

B˛k

This convergence will be justified later on. Furthermore and P 1N .y, 0/ D

1 ./ 1 ./

RC



00



N 1 

0

where

k

and

1

Vk P C .y, 0/ B˛k

RC

D

k ./ 1 ./

satisfy respectively

N 1 C .1 C bk .//˛ D 0 in .0, ˛k / 2 .˛k / D 0, lim  N 1 ./ D cN

(6.1.77)

N 1 0 N 1 C C D 0 in .0, 1/  2 ./ D 0, lim  N 1 ./ D cN .

(6.1.78)

C

!0

and 

00



lim

!1

!0

It is easy to verify that 1 ./

 .1N /=2  D cN  e .1 C o.1//

as  ! 1.

For any ı > 0 and > 0, there exists k,ı 2 N such that for any k  k,ı ,  2 Œı, ˛k  H) 1   .1 C bk .//˛  1 C .

(6.1.79)

214

Chapter 6 Further results on singularities and large solutions

The function  7! 00 k,



N 1  

k .=

0 k,

p 1 C / :D

N 1 C 2

k, ./

satisfies

p ˛ 1 C bk .= 1 C / k, C 1C 

D0 p in .0, ˛k 1 C /, k,

thus p p N 1 0 N 1 k, C k,  0 in . ı 1 C , ˛k 1 C /. k, C 2   p Since k, .ı 1 C / D k .ı/ remains uniformly bounded from below by p some constant Cı > 0 when k  k0 and 0 < p 1=2, and k, ../ C 1 .˛k 1 C / is a supersolution larger than 1 if  D ˛k 1 C , we finally derive   p p 8 2 Œı, ˛k . (6.1.80) 1 . 1 C /  1 .˛k 1 C / k ./  Cı 

00 k,



By (6.1.79), we see that there exists 0 > 0 such that  cN 1N  1N  2 e  .  2 e   1 ./  2cN 2 p We can always assume that ı 1 C  0 for 0 <  1=2, therefore

  0 H)

1 .

p

1 C / 

1 .˛k

p

1 C /

  p  p  1N  1N p p cN 2 2   1C ˛k 1 C e  1C  2cN e ˛k 1C . (6.1.81) 2 p p 2 A simple calculation shows that, if 0 = 1 C    ˛k  ln 8cN = 1 C , there holds   p  1N p p p cN 2  1C e  1C . 1 . 1 C /  1 .˛k 1 C /  4 p We fix  0 :D 15=49 (so that 1 C  8=7) and we obtain from (6.1.80)



8ı   0    ˛k  2 ln 4cN H) 7 Therefore, if

0 ˛k

 jxj  1 

uk .x/  vk .x/ D k˛kN 1

 2 ln 4cN ˛k

k ./

 Cı0 

1N 2

8

e 7 .

(6.1.82)

, it follows

k .˛k jxj/ 1

 x  8˛k jxj 1N  Cı0 k˛kN 1 .˛k jxj/ 2 e  7 . jxj (6.1.83)

215

Section 6.1 Singularities

Set ˇ.k, r / D k˛kN 1 .˛k r /

1N 2

e

8˛k r 7

ln ˇ.k, r / D ln k C

, then

8˛k r N 1 .ln ˛k  ln r /  . 2 7

Using (6.1.76) we obtain ln ˇ.k, r /  ln k 

8r ˛=2 ln k .1 C o.1// , 7

when k ! 1 and ˛k0  jxj  1  ˛2k . Since ˛  2 (it is at this point that we use the assumption on ˛), we obtain that ˇ.k, r / tends to infinity with k uniformly when  < r < 1  , for any 0 <  < 1=2. This implies that uk converges to 1 on any compact subset of B1C . If  is a ball, we can assume without loss of generality that it has radius 1=2 and center x0 D .0, ..., 1=2/. Let x1 D .0, ..., 1/ be the symmetric of 0 with respect to x0 and Ix1 the inversion with pole x1 , x 7! Ix1 .x/ D .x  x1/=jx  x1j2 . Then Ix1 B1=2 .x0/ is the half-space RN C tangent to B1=2 .x0 / at 0 which does not contain x0 . 2N If we set u.x/ D jyx1 j u.y/ Q with y D x1 CIx1 .x/. Therefore, if u is a solution of u C g ı u D 0 in B1=2 .x0/

(6.1.84)

for some real function g, then uQ satisfies Q D 0 in RN uQ C jy  x1 j2N g.jy  x1 jN 2 u.y// C.

(6.1.85)

Since DIx1 .0/ is the identity map, a standard calculation shows that if u D kı0 on @B1=2 .x0/ and thus (6.1.55) holds, then Z u.y/dS.y/ Q D cN k 8r > 0, (6.1.86) lim !0 † \Br

where † is the hyperplane ¹x D .0, 0, ..., 0, º and dS is the .N  1/-dimensional 2N : y 2 B C º and Lebesgue measure. Thus uQ D kı0 on @RN C . If a D max¹jy  x1 j 1 C N 2 b D max¹jy  x1 j : y 2 B1 º, then uQ C ag.b u/ Q  0 in B1C . If we assume in particular that g ı u D u ln ˛C u, and u D uk , then uQ D uQ k  vk in B1C , where vk is the solution of in B1C v C abv ln˛C.bv/ D 0 (6.1.87) v D kı0 on @B1C . If 0 < ˛  2, limk!1 vk D 1 locally uniformly in B1C . This implies easily that limk!1 vk D 1 locally uniformly in B1=2.x0 /. Finally, if  is any C 2 bounded domain and 0 2 @, there exist x0 2  and a ball B with center x0 tangent to @ at 0. By Lemma 6.1.8, uk,B  uk, . Since limk!1 uk,B D 1, uk, inherits the same property, locally uniformly in . 

216

Chapter 6 Further results on singularities and large solutions

Remark 6.5. The method used in the previous proposition can be adapted to deal with more superlinear nonlinearities than u ln ˛C u. For example, with u ln2C u .lnC lnC u/˛ , with 0 < ˛  2, the similarity factor ˛k has to be chosen in order to have  ˛ (6.1.88) ˛k2 D ln2 .k˛kN 1 / ln ln.k˛kN 1 / . The calculations become much more technical. An interesting question would be to find conditions on g 2 C.R/, g convex and increasing, besides the weak singularity assumption and K.a/ D 1 for any a > 0, in order to have u1 D 1. One of the striking consequences of Theorem 6.1.13 is the following: Theorem 6.1.14. Let  be a bounded C 2 domain in RN and 0  ˛  2. If u is a positive solution of u C u ln˛C u D 0 (6.1.89) in , then u ln ˛C u 2 L1 .; /. Furthermore u 2 L1./ and there exists a bounded positive Borel measure  on @ such that u D u , which is the solution of u C u ln ˛C u D 0 uD

in  on @.

(6.1.90)

Proof. If ˛ D 0 the result follows from classical harmonic analysis, thus we consider only ˛ > 0. Let us assume that u ln˛C u … L1 .; / and consider a sequence ¹ n º decreasing to 0 and set n D ¹x 2  : .x/ > n º. Then †n :D @n are uniformly C 2 hypersurfaces. We denote by 1,n the first eigenfunction of  in W01,2 .n / with corresponding eigenvalue 1,n (and drop the n for ) and with maximal value 1. Then Z Z Z @ 1,n udSn .x/. u 1,n dx C u ln˛C u 1,n dx D  (6.1.91) 1,n n n †n @n Furthermore there exists c > 0, independent of n, such that c†n .x/  1,n .x/  c 1 †n .x/

8x 2 n

and

@ 1,n .x/  c 1 8x 2 †n @n with corresponding relations with  and 1 . Since limn!1 1,n D 1 and †n .x/ " .x/, it follows from monotone convergence theorem that the left-hand side of (6.1.91) tends to infinity with n. Thus Z lim udSn .x/ D 1. c

n!1 † n

By compactness there exists x1 2 @ such that Z udSn.x/ D 1. lim n!1 † \B n 21 .x1 /

217

Section 6.2 Symmetries of large solutions

Iterating this process, we can construct a sequence of boundary points ¹xp º such that xpC1 2 B2p .xp / and Z udSn .x/ D 1 8p 2 N . lim n!1 † \B p .x / n p 2

Thus limp!1 xp D x  for some point x  2 @, and Z udSn .x/ D 1 lim n!1 † \B p .x  / n 2

8p 2 N .

(6.1.92)

Let k 2 N . Then for any p large enough, there exists np 2 N such that for any n  np , there exists mk,p,n 2 N such that Z min¹u, mk,p,n ºdSn.x/ D k, †n \B2p .x  /

and u  uk,p,n in n where uk,p,n is the solution of u C u ln˛C u D 0 u D min¹u, mk,p,n º

in n on †n .

(6.1.93)

If we take in particular n D np , the sequence ¹uk,p,np º converges to ukıx locally uniformly in  (see Chapter 3). Therefore u  ukıx . Since k is arbitrary and limk!1 ukıx D 1, it follows that u D 1, a contradiction. 

6.2 Symmetries of large solutions The moving planes method already used in the proof of Proposition 6.1.12 is at the core of the proof of the following classical result. Theorem 6.2.1. Let g : R 7! R be a locally Lipschitz continuous function, B  RN an open ball and u 2 C.B/ \ C 2 .B/ a positive solution of u C g ı u D 0 uD0

in B on @B.

(6.2.1)

Then u is radially symmetric with respect to the center of B. This result fails if g is no longer locally Lipschitz continuous. If u achieves a constant boundary value k, the result still holds if it is assumed either that u > k or u < k in B, which can be expressed by the fact that u is on one side of its boundary value. This is clearly the situation where u is a large solution in which case the following result holds.

218

Chapter 6 Further results on singularities and large solutions

Theorem 6.2.2. Let g : R 7! R be a locally Lipschitz continuous function, R > 0 and u 2 C 2 .BR / a positive solution of u C g ı u D 0 in BR lim u.x/ D 1.

(6.2.2)

jxj!R

If u satisfies lim ur .x/ D 1

jxj!R

hru.x/, i D o.ur .x//

as jxj ! R, 8 ? x,

(6.2.3)

then u is radially symmetric. Proof. Since the equation is invariant by rotation, it is sufficient to prove that u is C D BR \ ¹x D .x1 , x 0 / : x1 > 0º. symmetric in the x1 -direction, and we put BR C We claim first that, for any P 2 @BR there exists ı 2 .0, R/ such that ux1 .x/ > 0 8x 2 BR \ Bı .P /.

(6.2.4)

x x Since ru D ur jxj C r 0 u where r 0 u ? jxj satisfies jr 0 uj D o.ur / by (6.2.3), it follows   x1 x1 0 C hr u, e1 i D ur C o.1/ . (6.2.5) ux1 D ur jxj jxj

The moving plane method goes as follows: for 0 <  < R, let T be the hyperplane x1 D , † D ¹x 2 R :  < x1 < Rº and †0 D ¹x 2 R : 2  R < x1 < º. We also denote by x the point .2  x1 , x2, ..., xN / and by u the function x 7! u.x /. Let P0 D Re1 and ıP0 > 0 the ı defined in (6.2.4). If 0 D R 

2 ıP 0 , 2R

there holds

ux1 > 0 in † 0 [ †0 0 :D T 0 , therefore u.x / < u.x/

and ux1 > 0 in † ,

(6.2.6)

for   0. We define  as the infimum of the  > 0 such that (6.2.6) holds true, and we claim that  D 0. If it is not the case, we set K D T \ @BR . Since K is compact there exist > 0 and an open -neighborhood U of K such that ux1 .x/ > 0 8x 2 BR \ U .

(6.2.7)

We set D D BR=2 \ † and w D u  u . By definition w  0 in † and w C a.x/w D 0 in D where a.x/ D .g ı u  g.u //=.u  u / is bounded in D  . Therefore the maximum principle applies. If we take small enough and use the fact that u.x/ tends to infinity

219

Section 6.2 Symmetries of large solutions

when jxj ! R, it follows clearly that w is not identical to 0. It is therefore positive in D , and, since it vanishes on T , ux1 > 0 on T \@D . By continuity, this assertion jointly with (6.2.7), implies that there exists  > 0 such that ux1 .x/ > 0 8x 2 BR \ ¹x :    < x1 <  C º,

(6.2.8)

and we can choose > 0 small enough so that u > u in † . By definition of , there exist an increasing sequence ¹k º with limit  and points xk 2 † k such that u.xk /  u.xk k /. We can assume that xk ! xN 2 † . Since u.x/ N  u.xN  / if xN 2 † , but u > u in † , this implies that xN cannot belong to † . Furthermore, u.xk /  u.xk k / D 2.xk  k /ux1 .k / for some k in the open segment .xk , xk k /. If we assume that xN 2 T , then, for n large enough, the first coordinates .k /1 of k verify    < .k /1 <  C . Thus u.xk / > u.xk k / by using (6.2.8), which contradicts the definition of ¹xk º. We are left with x 2 @† n T . In that case u.xk /  u.xk k / would converge to 1 which again contradicts the definition of ¹xk º. Therefore  D 0, and in particular ux1 .0, x 0 /  0 for jx 0 j < R. Starting the reflection process from the point Re1 we obtain similarly ux1 .0, x 0 /  0. Therefore ux1 .0, x 0 / D 0 for any x 0 2 BR . Because of the invariance of the equation by rotation, it follows that hru.x/, i D 0 for all  x 2 BR and  ? x. This implies the result. Remark 6.6. Up until now the estimates (6.2.3) have not yet been obtained with a general nonlinearity g subject to the minimal required assumption for the existence of a large radial solution 9a 2 R s.t. g.r / > 0 8r  a

and K.a/ < 1,

(6.2.9)

where K.a/ is defined in (6.1.4). Sharper estimates can be obtained provided an asymptotic monotonicity is assumed, and more precisely that there exist M > 0 and two locally Lipschitz continuous functions gQ and g1 defined on R such that g D g1 C gQ

where gQ 0 on ŒM , 1/ and g1 is non-decreasing.

(6.2.10)

We denote by UR the maximal solution of U C g1 ı U D 0 in BR lim U.x/ D 1.

(6.2.11)

jxj!R

It is a radial function by construction. A proof of the fact that UR is the unique large solution of (6.2.11) under the mere monotonicity property has been recently obtained [31] and it is very technical. However, UR is the unique large solution except for at most a countable set of values R. This is a consequence of the fact that the mapping

220

Chapter 6 Further results on singularities and large solutions

which associates to a 2 RC the supremum R.a/ of the interval of existence of the maximal solution to the integral equation Z r Z s 1N s t N 1 g1 .ua .t //dt ds 8r 2 Œ0, R.a// (6.2.12) ua .r / D a C 0

0

is non-increasing. Therefore, the inverse mapping R.a/ 7! a, understood as a set value function, is also monotone. Therefore it is univalent except on a set X most countable values of R. The set X is the set of values of R for which uniqueness might not hold. Theorem 6.2.3. Assume that g is continuous and satisfies (6.2.9) and (6.2.10). If u 2 C 2 .BR / is a solution of (6.2.2) then it is radially symmetric. Lemma 6.2.4. Let the assumption of Theorem 6.2.3 be satisfied and u 2 C 2.BR / is a solution of (6.2.2). Then there exists K0 > 0 such that UR .x/  and

K0 2 K0 2 .R  jxj2 /  u.x/  UR .x/ C .R  jxj2 / 8x 2 BR . 2N 2N (6.2.13) Z jr.UR  u/j2 dx < 1. (6.2.14) BR

Furthermore there exists C > 0 such that jhru.x/, ij  C.R  jxj/

8x 2 BR

and  ? x with jj D 1.

(6.2.15)

Proof. Notice that we can always assume a D M in (6.2.9)–(6.2.10). Since u.x/ tends to infinity when jxj ! R, there exists 0 < R0 < R such that u.x/  M if jxj  R0 . Q : x 2 BR0 º. There holds Then g ı u D gQ ı u 1BR C g1 ı u. Put K0 D sup¹jg.u.x//j 0

K0  u C g1 ı u  K0 . 1

(6.2.16)

Set .x/ D .2N / .R  jxj /; then  D 1 in BR and vanishes on @BR . Therefore .u  K0 /  g1 ı u  .u C K0 / 2

2

since g1 is non-decreasing. Therefore uK0 is a subsolution of the problem (6.2.11). It is smaller than the maximal solution UR . Similarly, using approximation of UR by maximal solution UR0 of (5.2.18) in BR0 with R0 > R and the uniqueness of large solution, we obtain that u C K0  UR . Thus (6.2.13) holds. Next we consider w D UR  u. It satisfies w C d.x/w D 0 in BR n BR0 , where d.x/  0 since

8 < g1 ı UR  g1 ı u dD UR  u :0

if UR .x/ ¤ u.x/ if UR .x/ D u.x/.

(6.2.17)

221

Section 6.2 Symmetries of large solutions

Because of (6.2.13), for any > 0 the function .w  /C vanishes in a neighborhood of @BR and d.x/w.w  /C  0, thus Z Z 2 jr.w  /C j dx  wr .w  /CdS  C1. BR nBR0

@BR0

Letting ! 0 and using the fact that w 2 C 1 .B R0 / yields Z jrwC j2 dx < 1. BR

Since .w  / also has compact support, we obtain (6.2.14). Let A be a skew-symmetric matrix and ¹e tA º t 2R the 1-parameter group of orthogonal transformations that it generates. Since the equation is invariant with respect to such a transformation, x 7! u.e tA x/ :D u ı e tA .x/ satisfies (6.2.2) and is larger than M in BR n BR0 . Because UR and are radially symmetric, there holds K0 2 K0 2 .R  jxj2 /  u.e tA x/  UR .x/ C .R  jxj2 / 8x 2 BR . 2N 2N (6.2.18) Combining (6.2.16) and (6.2.18), we derive then for all t 2 R and x 2 BR , UR .x/ 

2

K0 2 K0 2 .R  jxj2 /  u.e tA x/  u.x/  2 .R  jxj2 /. 2N 2N

(6.2.19)

Set W t .x/ D u.e tA x/  u.x/, then W t C d t .x/W t D 0 in BR n B R0

(6.2.20)

where d t  0 since

8 < g1 ı u ı e tA  g1 ı u dt D u ı e tA  u : 0

if u.e tA x/ ¤ u.x/ if u.e tA x/ D u.x/.

By the chain rule and the Taylor expansion W t .x/ D t hru.e  tA x/, Ae tA xi, for some  2 .0, 1/. Notice also that if jxj D R0 then je tA xj D R0 . Since u 2 C 1 .BR /, jW t .x/j  jt jR0 kAk sup ¹kru.z/k : jzj D R0 º D C jt j. Put

8 2N  R2N jxj ˆ ˆ ˆ < R2N  R2N 0 P .x/ D ˆ ˆ ln jxj  ln R ˆ : ln R0  ln R

if N  3 if N D 2.

222

Chapter 6 Further results on singularities and large solutions

Therefore C jt jP ../ is a supersolution for (6.2.20). Since W t .x/ ! 0 when jxj ! R, for any > 0 the function .W t  C jt jP  /C is a subsolution with compact support. Therefore, W t  C jt jP C , and W t  C jt jP by letting ! 0. Similarly W t  C jt jP . Dividing by t and letting t ! 0 yields jhru.x/, Axij  kAk jxj sup ¹kru.z/k : jzj D R0 º P .x/

in BR n B R0 . (6.2.21)

We infer that (6.2.15) holds since Ax ? x and the space of symmetric matrices acts transitively on RN .  Remark 6.7. Estimate (6.2.13) admits an extension to a more general smooth bounded ı

domain  such that  D  (see Chapter 4). If we assume that g satisfies (6.2.9)– (6.2.10) and that the maximal solution of U C g1 .U / D 0 in  is the unique large solution of this equation, then any large solution u of u C g ı u D 0 in ,

(6.2.22)

which is larger than M in 0r0 :D ¹x 2  : .x/ < r0º, satisfies UR .x/  K0 .x/  u.x/  UR .x/ C K0 .x/

8x 2 ,

(6.2.23)

Q : x 2 n0r0 º and satisfies  D 1 in  and vanishes where K0 D max¹g.u.x// on @. Estimate (6.2.14) is also valid. Lemma 6.2.5. Let the assumption of Theorem 6.2.3 be satisfied and u 2 C 2 .BR / be a solution of (6.2.2) larger than M on BR n BR0 . Then there exist two radial functions ui (i D 1, 2) satisfying ui C g1 ı ui D 0 in BR n B R0 lim ui .x/ D 1,

(6.2.24)

jxj!R

and UR  K0  u2  u  u1  UR C K0

in BR n B R0 ,

and a constant C > 0 such that Z R jg1 .u.r , //  g1 .ui .r , //j r N 1 dr  C R0

8 2 S N 1 .

(6.2.25)

(6.2.26)

Proof. Step 1. Let ui 2 C 2 .BR n BR0 / (i D 1, 2) be two radial functions verifying (6.2.24) such that u1  u2 and limjxj!R .u1  u2 /.x/ D 0. We claim that Z

R R0

.g1 ı u1  g1 ı u2 /dx < 1.

(6.2.27)

223

Section 6.2 Symmetries of large solutions

When N > 2 we consider the change of variable s D r 2N , ui .r / D vi .s/. Then N 1

vi00 D cN s 2 N 2 g1 ı vi

on .S, S0 /

(6.2.28)

where S D R2N and S0 is defined accordingly. Since g1 is non-decreasing and v1  v2 , there holds N 1

.v1  v2 /00 .s/ D cN s 2 N 2 .g1 ı v1  g1 ı v2 /  0.

(6.2.29)

Thus v1 v2 is convex and positive on .S, S0 / with lims!S .v1 v2 /.s/ D 0. Therefore v1  v2 is increasing and there exists c 2 Œ0, 1/ such that lim

s!S

.v1  v2 /.s/ D c D lim .v10  v20 /.s/. sS s!S

Integrating (6.2.29) we get .v10  v20 /.S0 /  c D cN

Z

S0 S

N 1

s 2 N 2 .g1 ı v1  g1 ı v2 /ds.

This reads Z .N  2/cN

R R0

.g1 ı u1  g1 ı u2 /r N 1 dr  .v10  v20 /.S0 /  c.

(6.2.30)

If N D 2, the change of variable s D ln r , ui .r / D vi .s/ leads similarly to inequality Z R c .g1 ı u1  g1 ı u2 /rdr  .v10  v20 /.S0 /  c. (6.2.31) R0

Step 2. If u is a solution of (6.2.2), we set U1 .r / D max¹u.r , / :  2 S N 1 º and U2 .r / D min¹u.r , / :  2 S N 1 º. By a standard convex analysis argument, the following inequality holds in the sense of distributions in .R0 , R/ U100 C

N 1 0 U1  max¹g.u.r , // :  2 S N 1 º D g1 ı U1 . r

Similarly U200 C

N 1 0 U2  min¹g.u.r , // :  2 S N 1 º D g1 ı U2 r

in D 0 .R0 , R/.

Furthermore, by (6.2.13), UR  K0  U2  u  U1  UR C K0

in BR .

Therefore, there exist two radial functions u1 and u2 satisfying ui C g1 ı ui D 0 in BR n B R0

(6.2.32)

224

Chapter 6 Further results on singularities and large solutions

and UR  K0  u2  U2  u  U1  u1  UR C K0

in BR n B R0 .

(6.2.33)

Because u1  u2 and limjxj!R .u2 .jxj/  u1 .jxj// D 0, we derive Z

R R0

.g1 ı u1  g1 ı u2 /r N 1 dr < 1.

Finally (6.2.26) is inferred from (6.2.25) and the monotonicity of g1 .



Proof of Theorem 6.2.3. Let ui (i D 1, 2) be the functions defined in the previous lemma. We extend them on Œ0, R0 / by i where i 2 C 2 .Œ0, R0 / satisfies i0 .0/ D 0, i .R0 / D ui .R0 /, i0 .R0 / D u0i .R0 / and i00 .R0 / D u00i .R0 /, and we denote by uQ i the resulting functions defined on Œ0, R/. Considered as functions defined in BR , they are radial solutions of uQ i C fQi D 0 in BR where fQi 2 C.BR / coincides with g1 ı uQ i D g1 ıui in BR nBR0 . We set w1 D uQ 1 u; then w1 is non-negative in BR n BR0 and w1.x/ ! 0 when jxj ! R. Furthermore fQ1  g ı u 2 L1.BR / \ C.BR / by (6.2.26). It follows that w1 is the unique solution of w1 C fQ1  g ı u D 0 w1 D 0 Therefore w1 satisfies

Z

w1 .x/ D

BR

in BR on @BR .

  GR .x, y/ fQ1 .y/  g.u.y// dy,

(6.2.34)

(6.2.35)

where GR .x, y/, the Green kernel in BR , can be written as GR .x, y/ D cN jx  yj2N C .x, y/ (with the usual modification if N D 2) and 2 C 2.B R /. We have Z   @w1 D .2  N /cN jx  yj1N fQ1 .y/  g.u.y// dy @r BR Z  x  Q i f1 .y/  g.u.y// dy hrx , C jxj BR

(6.2.36)

D A1 C A2 . x The second term in the right-hand side of (6.2.36) is easy to estimate, since hrx , jxj i is uniformly bounded, therefore Z Z Q jA2 j  C jf1  g ı ujdy C jg1 ı u1  g1 ı ujdy  C 0 , BR0

BR nBR0

225

Section 6.2 Symmetries of large solutions

by Lemma 6.2.5. For the first term, we use the fact that Z jx  yj1N jfQ1 .y/  g.u.y//jdy BR

Z D Z

R

@Br

0 R



Z

jx  yj1N jfQ1 .y/  g.u.y//jdS.y/dr

° ±Z Q max jf1 .y/  g.u.y//j : jyj D r

0

@Br

jx  yj1N dS.y/dr .

If we write y D .y1 , y 0 / 2 R RN 1 and assume that x D .x1 , 0, ..., 0/ with x1 2 Œ0, 1/ (which can be always assumed up to a rotation), we have Z rZ Z 1N 1N jx  yj dS.y/ D ..x1  y1 /2 C jy 0 j2 / 2 dS.y 0/dy1 p 2 @Br r jy 0 jD r 2 y1 Z r 1N N 2 D N .x12 C r 2  2x1y1 / 2 .r 2  y12 / 2 dy1 r 1

Z D N

1

.X12 C 1  2X1 z/

1N 2

.1  z 2 /

N 2 2

dz,

:D I.X1 /, where we have put X1 D r 1x1 2 Œ0, 1/. If 0  X1 < 1, X12 C 1  2X1 z > 0 for any z 2 Œ1, 1, thus X1 7! I.X1 / is continuous on Œ0, 1/. Furthermore Z 1 1N 1N N 2 2

N .1  z/ 2 .1  z 2 / 2 dz < 1 I.1/ D 2 1

and I is continuous at X1 D 1. Therefore there exists M > 0 such that Z jx  yj1N jfQ1 .y/  g.u.y//jdy BR

Z

R

M

° ± max jfQ1 .y/  g.u.y//j : jyj D r dr , (6.2.37)

0 1 is uniformly and the right-hand side of (6.2.37) is also bounded. Therefore @w @r 0 bounded. But limr !R u1 .r / D 1: this is a consequence of equation (6.2.28), convexity and the fact that v1 .s/ ! 1 as s ! S. Therefore

@u D 1. r !R @r lim

Combining this estimate with (6.2.15) and using (6.2.3) we obtain the claim.



226

Chapter 6 Further results on singularities and large solutions

6.3 Sharp blow up rate of large solutions Let   RN be a bounded domain and g : R 7! R a continuous non-decreasing function positive on .0, 1/ satisfying the positive Keller–Osserman condition. A natural question, linked to the uniqueness consideration, is to express the boundary behavior of any function which satisfies u C g ı u D 0 in  lim u.x/ D 1

(6.3.1)

.x/!0

as a function of .x/ D dist .x, @/. We denote by ˆ the solution of ˆ00 C g ı ˆ D 0 in .0, 1/ lim .t / D 1.

(6.3.2)

t !0

It is expressed by

Z tD

where G.t / D satisfies

Rt 0

1 ˆ.t /

ds p :D ‰.ˆ.t //, 2G.s/

(6.3.3)

g.s/ds. We have seen in Theorem 4.1.2 that any solution of (6.3.1) lim sup .x/!0

‰.u.x//  1. .x/

(6.3.4)

If  is C 2 , this inequality is improved independently of the geometry of @ since there holds ‰.u.x// lim D 1. (6.3.5) .x/!0 .x/ A natural question is to find conditions under which (6.3.5) can be made more precise. If there holds ‰.ˇt / > 1 8ˇ 2 .0, 1/, (6.3.6) lim inf t !1 ‰.t / the estimate follows u.x/ D 1. .x/!0 ˆ..x/ lim

(6.3.7)

Jointly with the convexity of g it implies the uniqueness of large solutions. A second type of improvement is to see under what condition there holds lim .u.x/  ˆ..x// D 0.

.x/!0

We will see that this estimate depends on the curvature of @.

(6.3.8)

227

Section 6.3 Sharp blow up rate of large solutions

6.3.1 Estimates in an annulus The starting point for obtaining sharp two-sided estimates of the blow up rate of solution of (6.3.1) in a C 2 domain is to study solutions in an annulus. Proposition 6.3.1. Let  D ¹x 2 RN : R0 < jxj < R1 º and Z tp 2G.s/ds .t / D 0 . G.t / If u is a radial solution of u C g ı u D 0 in , the following implications hold (i) If limr "R1 u.r / D 1, then   Z R1 N 1 .1 C o.1// .ˆ.s//ds u.r / D ˆ R1  r  2R1 r

(6.3.9)

(6.3.10)

as r ! R1 .

(6.3.11) This formula is valid even if R0 D 0 and u is a solution in BR1 . (ii) If limr #R0 u.r / D 1, then   Z r N 1 u.r / D ˆ r  R0 C .1 C o.1// .ˆ.s//ds as r ! R0 . 2R0 R0 (6.3.12) c This formula is valid even if R1 D 1 and u is a solution in BR0 .

Proof. We give the proof of (i), the one of (ii) being similar. We multiply (6.3.10) by u0 , integrate on Œr0 , r   .R0 , R1 / and get Z r 02 u .s/ u02 .r / u02 .r0 /  C .N  1/ ds D G.u.r //  G.u.r0 //. 2 2 s r0 Equivalently

  h.u.r0 // u .r / C 2.N  1/I D 2G.u.r // 1 C , G.u.r // 02

where

Z I D

r r0

(6.3.13)

u02 .s/ u02 .r0 / ds and h.u.r0 // D  G.u.r0 //. s 2

It implies



 h.u.r0 // u .r /  2G.u.r // 1 C . G.u.r // that u0 > 0 It is clear from the maximum principle that there exists rN 2 .R0 , R1 / such p 00 on ŒNr , R1 /, therefore, choosing r0 2 ŒNr , R1 /, we have u  g ı u  c G.u/ on 02

228

Chapter 6 Further results on singularities and large solutions

Œr0, R1 /, for some c > 0. The assumption on g implies that u is convex near R1 and u0 is increasing. Up to replacing r0 by a larger value,   r 02 D o.u02 .r // as r ! r0. I  u .r / ln r0 Plugging this estimate into (6.3.13) and using the fact that g.t / D o.G.t // when t ! 1, we obtain u02 .r / D 2G.u.r // .1 C o.1//

as r ! R1 .

It yields the estimate Z Z 1 C o.1/ r p 1 C o.1/ r 02 u ds D 2G.u/u0 ds I D R1 R1 r0 r0 1 C o.1/ D R1

Z

(6.3.14)

u.r / p

2G.t /dt .

u.r0 /

Therefore

I .u.r // D .1 C o.1// as r ! R1 . G.u/ R1 Using the fact that .t / D o.1/ as t ! 1, (6.3.13) becomes  1=2 p .N  1/.u.r // u0 .r / D 2G.u.r // 1  .1 C o.1// R1 (6.3.15)   p .N  1/.u.r // D 2G.u.r // 1  .1 C o.1// . 2R1 p If we divide this expression by 2G.u.r // and integrate on Œr0 , r , we obtain Z N 1 r ‰.u.r0 //  ‰.u.r // D r  r0  .1 C o.1// .u.s//ds. 2R1 r0 Letting r ! R1 and since ˆ D ‰ 1 , we conclude that (6.3.12) holds.



Corollary 6.3.2. Assume  is a bounded C 2 domain. Then any function u verifying (6.3.1) satisfies (6.3.5). Proof. Let R > 0 such that at any z 2 @, the ball BR .a/ is interior to  and tangent at @ at z and the ball BR .b/ is exterior to  and tangent at @ at z. Assume also that the diameter of  is smaller than R0 CR for some R0 > 0. If x 2  is such that .x/ < R, we denote by z D zx a projection of x onto @ and by a D ax and b D bx the centers of the corresponding interior and exterior spheres tangent at @ at z. Let ui and ue be the large solutions of (6.1.61) respectively in BR .a/ and BR0 CR .b/ n Bb .R/. Then (i) (ii)

u.x/  ui .x/ ue .x/  u.x/

in BR .a/ in .

(6.3.16)

229

Section 6.3 Sharp blow up rate of large solutions

It follows from (6.3.11)–(6.3.12) N 1 ‰.ui .x// D .x/  .1 C o.1// 2R

Z

R

.ˆ.s//ds R.x/

as .x/ ! 0 (6.3.17)

and ‰.ue .x// D .x/ C

N 1 .1 C o.1// 2R

Z

RC.x/

.ˆ.s//ds R

Since ‰ is monotone (6.3.5) follows.

as .x/ ! 0. (6.3.18) 

Remark 6.8. In these expressions, the two terms R11 and R01 are the curvatures of the boundary of the domain. The explicit role of the mean curvature in the expression of secondary effects for large solutions will be made more explicit in next section. Remark 6.9. For an alternative expression of Proposition 6.3.1, we introduce Z R1 N 1 !1 .r / D .1 C o.1// .u.s//ds 2R1 r Z r N 1 .1 C o.1// .u.s//ds. 2R1 R0 By using the mean value theorem, (6.3.11) reads as follows

and

!0 .r / D

u.r / D ˆ.  !1 .r // D ˆ./  ˆ0 .1 /!1 .r /

(6.3.19)

where  D R1  r  1    !1 .r /, and (6.3.18) as u.r / D ˆ. C !0 .r // D ˆ./ C ˆ0 .0 /!0 .r /

(6.3.20)

with  D r  R0  0   C !0 .r /. An important consequence of Proposition 6.3.1 is the following Proposition 6.3.3. Let the assumptions of Proposition 6.3.1 be verified. Assume that t 7! t 2 G.t / is increasing at infinity and let u be a radial solution of (6.3.10) in . (i) If limr "R1 u.r / D 1, then ˆ./  u.r /  ˆ./ C .1 C o.1//

.N  1/ ˆ./ as  D R1  r ! 0. R1 (6.3.21)

(ii) If limr #R0 u.r / D 1, then ˆ./  u.r /  ˆ./  .1 C o.1//

.N  1/ ˆ./ as  D r  R0 ! 0. R0 (6.3.22)

230

Chapter 6 Further results on singularities and large solutions

Proof. We carry out the proof of (i), the proof of (ii) being similar. Using the expression of  and the monotonicity of G, we get p 2 u.r / .u.r //  p . G.u.r // If u is increasing near R1 and since t 7! t 2 G.t / is increasing at infinity, p p Z R1 2 u.r / 2 u.r / .u.s//ds  p .R1  r / D p , G.u.r / G.u.r / r

(6.3.23)

on some Œr0 , R1 /. By (6.3.19) and since ˆ0  0, p p p ˆ0 .1/ D 2G.ˆ.1//  2G.ˆ.  !1 .r /// D 2G.ˆ.u.r //. Inserting this estimate into (6.3.19) and using the expression of !1 and (6.3.23), we derive u.r /  ˆ./ C Then

p N 1 2G.u.r // !1  ˆ./ C .1 C o.1//u.r /. R1

(6.3.24)

  N 1 .1 C o.1// u.r /  ˆ./, 1 R1

and this estimate implies the right-hand side inequality in (6.3.21). The left-hand side follows from (6.3.19).  If F has stronger growth at infinity, the previous estimates can be improved. Corollary 6.3.4. Under the assumptions of Proposition 6.3.3 and the additional assumption that t 4 G.t / ! 1 when t ! 1, there holds lim .u.r /  ˆ.// D 0,

!0

(6.3.25)

where  D R1  r or r  R0 , accordingly u satisfies (i) or (ii). Proof. It is an immediate consequence of a2 D 0. lim ˆ./ D lim a‰.a/ D lim p a!1 a!1 !0 2G.a/



Theorem 6.3.5. Let  be a bounded domain with a boundary satisfying the inner and outer sphere condition. Assume that g : R 7! R is continuous, non-decreasing such that g.0/  0 and satisfies the positive Keller–Osserman condition. If t 7! t 2 G.t / is non-decreasing at infinity, then

231

Section 6.3 Sharp blow up rate of large solutions

(i) there exists a constant c > 0 such that every solution of (6.3.1) satisfies ˇ ˇ ˇ u.x/ ˇ ˇ ˇ  c.x/ 8x 2 ;  1 (6.3.26) ˇ ˆ..x// ˇ (ii) if in addition t 4 G.t / ! 1 when t ! 1, then every solution of (6.3.1) verifies lim .u.x/  ˆ..x/// D 0.

.x/!0

(6.3.27)

Proof. Let R1 > 0 such that for any z 2 @ there is a ball BR1   such that @ \ B R1 D ¹zº. Further let A be an annulus of outer radius R1 , inner radius R0 , center a, such that its inner boundary ¹x : jx  aj D R0 º lies outside , but jz  aj D R0 . Let v be a radial large solution of (6.3.1) in BR and w a radially symmetric solution of (6.3.1) in A such that w.x/ D 0 if jx  aj D R1 and limjxaj!R0 w.x/ D 1. By comparison and approximation u  v in BR and w  u in A \ . The conclusion follows from Proposition 6.3.3 and Corollary 6.3.4. 

6.3.2 Curvature secondary effects If g.r / D r q (q > 1) and @ is C 2 , one can construct radial super- and subsolutions of (6.3.1) in interior balls or exterior spherically symmetric domains and prove that any large solution u satisfies   N 1 u.x/ D ˆ..x// 1 C H0..x//.x/ C o..x// as .x/ ! 0, qC3 (6.3.28) where .x/ is the projection of x onto @ (uniquely defined if .x/ is small enough) and H0 .a/ is the mean curvature of @ at a. Notice that, in that case,  ˆ.t / D

2.q C 1/ .q  1/2 t 2



1 q1

.

(6.3.29)

If g.r / D e r it is proved that   2 u.x/ D ln C .N  1/H0 ..x//.x/ C o..x// as .x/ ! 0. (6.3.30)  2.x/ The curvature secondary effects have been thoroughly explored in [12] with more general nonlinearities. There is a need to introduce g.t / B.t / D p 2G.t / Z N 1 t J.t / D .ˆ.s//ds, 2 0

(6.3.31)

232

Chapter 6 Further results on singularities and large solutions

where  is defined above and consider two additional technical assumptions (i) (ii)

lim

ı !0

B .ˆ.ı.1 C oı//// D1 B.ˆ.ı//

(6.3.32)

lim sup B.ı/.ı/ < 1. ı !1

Actually, only the following consequence of (6.3.32) (ii) is used: for any  > 0 there exists M > 0 such that J./ 

N 1 2 2 M 

8 2 .0, .

(6.3.33)

The next result points out the role of the curvature in the boundary behavior of solutions of (6.3.1). Theorem 6.3.6. Assume that  is a bounded domain with a C 4 boundary and g : R 7! R is continuous, non-decreasing, positive on .0, 1/ and satisfies the positive Keller–Osserman condition. Assume also that (6.3.6) and (6.3.32) hold, t 7! t 2 G.t / is asymptotically non-decreasing and there exists ˛ 2 .0, 1/ such that ˆ0 .˛ı/ < 1. 0 0<ı 1 ˆ .ı/ sup

(6.3.34)

Then any solution u of (6.3.1) satisfies   u.x/  ˆ .x/  J..x//H0 ..x// D ˆ..x//o..x// as .x/ ! 0. (6.3.35) Remark 6.10. Notice that the Keller–Osserman condition implies that lim t !1 .t / D 0 and therefore J..x// D o..x// when .x/ ! 0, but condition (6.3.32) (i) implies a stronger decay, actually J..x/ D O. 2.x//. The proof is very technical and we shall just sketch it. It necessitates the use of standard geometric tools associated to the curvature of an hypersurface. If D is a bounded domain with a C 4 boundary †, we set .x/ D dist .x, †/ and, for ı > 0, Dı D ¹x 2  : .x/ < ıº and †ı D ¹x 2  : .x/ D ıº. If  2 †, we denote by .1 , ..., N 1 / its coordinates in some local chart and by x./ D .x1./, ..., xN .// its cartesian coordinates. If † is C k (k  2), there exists 0 > 0 such that for any x 2 D0 there exist a unique projection .x/ of x onto †. Furthermore, the mapping x 7! ..x/, .x// is a C k  1 diffeomorphism of D0 onto † .0, 0/. Finally N C n, 8x 2 D0 , ….x/ D ., / H) x D x./

233

Section 6.3 Sharp blow up rate of large solutions

where n D n is the normal inward unit vector at . If the coordinates are taken with respect to the curves defined by the vector field of the principal curvatures N i :D N i ./ on †, then the metric tensor gN is diagonal. In this system of coordinates the Laplacian is expressed in the following way u D  u C  u, where

1 X @  u D p jgj i ,j @j



p

g

@u 2 2 @ gN ij .1  2 N i  C N i  / i

 (6.3.36)

@2 u @u  u D 2  .N  1/H , @ @ (with jgj :D det g) and H., / :D

1 X N i . N 1 1  N i 

(6.3.37)

i

Notice that H., / is the mean curvature of † , and H0./ D

1 X N i ./ N 1 i

is the mean curvature of †. The key tool of the proof of Theorem 6.3.6 relies on the construction of local super- and subsolutions in a neighborhood of @. The expression of these auxiliary functions contain the mean curvature. Proposition 6.3.7. Assume that the positive Keller–Osserman condition and (6.3.32) hold. Then there exist 0 > 0 and 0 > 0 such that for any  2 .0, 0  there exists C and   defined in D  2 .0, 0  such that the two functions   :D .0,   † by    Z  N 1 ˙ .H0 ./ ˙ /  ., / D ˆ   .ˆ.s//ds (6.3.38) 2 0 are respectively a supersolution and a subsolution of (6.3.10) in D . Abridged proof. Set

  Z  N 1 ˙ .H0 ./ ˙ / ., / D ˆ   .ˆ.s//ds .  2 0

(6.3.39)

Since ˆ is non-decreasing, there exist 0 > 0 and 0 > 0 such that if 0    0 there holds Z  N 1 .H0 ./ C / > .ˆ.s//ds 8., / 2 .0, 0  †, 2 0

234

Chapter 6 Further results on singularities and large solutions

C  and thus  ., /   ., /. Given  2 R, we set ., / D   .H0 ./ C /J./ (see (6.3.31) for the expression of J ), therefore  D ˆ ı  (we drop the exponent ˙ since we do not impose any sign on , only the restriction jj  0 ). Then

jrj D 2

N 1 X

2i

i D1

gN i i .1  2 N i  C N i2 2 /

and since i D J./H0 i ./ and  D 1 

N 1 .H0 ./ 2

,

C /.ˆ.//,

  D J./ H0 ./, N 1 .H0 ./ C / 0 .ˆ.//ˆ0./   D  2   N 1  .N  1/H., / 1  .H0 ./ C /.ˆ.// , 2 p where H is defined in (6.3.37). Since ˆ0 D  2G ı ˆ there holds  0 .ˆ.//ˆ0 ./ D 2.1 C B.ˆ/.ˆ//. Using the fact that H., /  H0 ./ D O./, we obtain   D .N  1/ .  .H0 ./ C /B.ˆ/.ˆ// C O./.

(6.3.40)

Because @ is C 4 , one can verify that there exist two bounded functions k1 and k2 in D0 such that   D J./k1 ., / and jrj2 D J./k2 ., /. Since  D ˆ0 ./ C ˆ00 ./jr 2 j and ˆ00 ./ D .g ı ˆ/.// D g. /, we derive from the previous relations  D g. / C ˆ0 ./ .N  1/  B. /J 2 ./k2

C .N  1/.H0 ./ C /.ˆ/.B. /  B.ˆ// C O./ . (6.3.41) Since  ., / D ˆ..1 C o.1///, assumption (6.3.13) implies that lim ŒB ., //  B.ˆ.//.ˆ.// D 0.

!0

Using (6.3.33) we obtain that B. ., // D o. 2/. Jointly with (6.3.41) it yields the statements.  Abridged proof of Theorem 6.3.6. By Proposition 6.3.7, for every  2 .0, 0  and  2  C .0,  , there exists a solution u D u of (6.3.10) in D such that   u   . By Theorem 6.3.5 (i) ju.x/  ˆ..x//j  c.x/ˆ..x//  c 0 ˆ0 ..x//J..x//.

235

Section 6.4 Notes and comments

By (6.3.38) and the mean value theorem C D ˆ.  H0J.//  ˆ0 .O C/J./ D ˆ./  ˆ0 .Q C /.H0 C J.//,  (6.3.42)

and  D ˆ.  H0J.// C ˆ0 .O  /J./ D ˆ./  ˆ0 .Q /.H0  J.//, (6.3.43) 

where O ˙ is between   H0J./ and   .H0 ˙ /J./, and Q ˙ between   H0J./ and   .H0 ˙ /J./. Thus O ˙ and Q ˙ are of order  C o./. Using assumption (6.3.34), we derive from (6.3.42)–(6.3.43) ˇ ˙ ˇ ˇ .x/  u.x/ˇ  cˆ0 ..x//J..x// 8x D ., / 2 D . (6.3.44) 

Jointly with (6.3.38) and the definition of J , this implies   u.x/  ˆ .x/  .H0 ./ C /J..x//  cˆ0 ..x//J..x//. Since  can be arbitrarily small, the upper estimate in (6.3.35) follows from the as sumptions on ˆ0 . The proof of the lower estimate is similar. Estimate (ii) in Theorem 6.3.5 can also be improved. Corollary 6.3.8. Under the assumptions of Theorem 6.3.6, if we suppose moreover t4 < 1, t !1 G.t / lim

there holds u.x/ D ˆ..x// C ˆ0 ..x//H0 ./J..x// C o.1/ as .x/ ! 0.

(6.3.45)

6.4 Notes and comments In [103], Y. Richard and L. Véron introduced a sweeping method for studying the isolated singularities of solutions of u C u .ln uC /˛ D 0,

(6.4.1)

since the Harnack inequality was inefficient to prove the isotropy of the blow up. In particular, they proved Proposition 6.1.3 and Theorem 6.1.2, from which the complete classification of isolated singularities is derived. The study of isolated boundary singularities for positive solutions of (6.4.1) was initiated in [47]. In this paper the case ˛ > 2 is treated completely while the case 0 < ˛  2 remained partially open; Theorem 6.1.13 provides a simpler and complete proof of the results therein. Some similar types of problems, but for the parabolic equation u t  u C u .ln uC /˛ D 0,

(6.4.2)

236

Chapter 6 Further results on singularities and large solutions

are thoroughly treated in [94]. In particular, the importance is pointed out of the stability when approximating the fundamental solutions uk of these equations for describing the boundary trace (or the initial trace) of positive solutions (as far as (6.4.2) is concerned, uk is a solution with initial data kı0 ). An analog of Theorem 6.1.14 concerning (6.4.2) is proved in [94]. An alternative approach for Proposition 6.1.1 in the case II (when  D BR ) is to consider the Cauchy problem u00 C Nr1 u0  g ı u D 0 u.R/ D 0, u0 .R/ D ˛,

in .0, R/

(6.4.3)

with a solution u :D u˛ expressed by ˛RN 1 2N 1 .r u˛ D  R2N / C 2N N 2

Z

R

r

.r 2N  s 2N /f .u˛ .s//ds. (6.4.4)

If ˛ < 0 the function u˛ is positive and decreasing on the maximal interval on the left of R, and if K.a/ D 1 for any a > 0, it can be continued up to 0. Furthermore ˛ < ˛ 0 < 0 H) u˛ > u˛0 . For any k > 0 the solution uk of (6.1.1) in  D BR is of the form u˛k for some unique ˛k < 0 and k > k 0 H) ˛k < ˛k 0 < 0. Therefore, if limk!1 ˛k D ˛  > 1, there exist infinitely many positive solutions u of (6.1.8) in BR n ¹0º vanishing on @BR and satisfying limx!0 jxjN 2 u.x/ D 1,

(6.4.5)

and any solution u˛ with ˛ < ˛  cannot be a -moderate solution of (6.1.8) in BR n ¹0º. Conversely, if limk!1 ˛k D 1, then limk!1 uk D 1. It is an open problem to prove the existence of functions g 2 G0 for which limk!1 ˛k > 1. The question of symmetry of solutions of nonlinear elliptic equations in a ball is connected to the general question whether the invariance of a domain G under some subgroup of orthogonal transformations implies the invariance of solutions of u C g ı u D 0 uDk

in G, on @G

(6.4.6)

without relying on a uniqueness result. In this formulation k may be finite or infinite and G bounded or unbounded. The first striking result is due to W. M. Ni, B. Gidas and L. Nirenberg [52] who proved that if G is a ball, f is locally Lipschitz continuous and u is a solution of (6.4.6) such that u  k has a constant sign, then u is radial. Many extensions of this result followed. Since a large solution u is always on one side of its boundary value, it was conjectured by Brezis that large solutions in a ball are radial. Theorem 6.2.2 was proved by A. Porretta and L. Véron in [99] who show that the assumptions always hold when f is asymptotically convex. In 2008, O. Costin and L. Dupaigne [31] proved that if f is asymptotically monotone, then Theorem 6.2.2 holds.

Section 6.4 Notes and comments

237

They also proved that if f is monotone, f .0/  0 and satisfies the positive Keller– Osserman condition, then (6.4.6) where G is a ball and k D 1 admits a unique radial solution. A. C. Lazer and P. J. McKenna [64] raised the question: under what condition on g estimate (6.3.8) holds. Clearly this estimate represents a strong improvement over previous estimates of [9, 114] such as (6.3.7). They showed that under some conditions on g and on the domain the answer is affirmative. A more precise result, namely Proposition 6.3.1, was obtained by C. Bandle and M. Marcus [9]. Applied to the special case of power nonlinearities their result shows that if q > 3 then ˇ ˇ ˇ ˇ (6.4.7) ˇu.x/  k1,q ..x//2=.q1/ ˇ  c..x//.q3/=.q1/ . S. Berhanu and G. Porru [16] concentrated on the case of power nonlinearities. Using the methods of [9] they obtained, for 1 < q < 3, a full expansion ˇ ˇ ˇ ˇ m X ˇ ˇ j 2=.q1/ ˇ m2=.q1/ ˇu.x/  aj ..x// (6.4.8) ˇ  c..x// ˇ ˇ ˇ j D0 where m is an integer such that 0 < m C 1  2=.q  1/  1 and the aj depends on  if j > 0. In another special case, namely, q D .N C 2/=.N  2/, L. Anderson, P. T. Chrusciel and H. Helmut obtained in [6] an infinite expansion of the solution in terms of some geometric functions associated with the boundary. However, an explicit estimate of this terms does not seem to be available. Bandle and Marcus [11] obtained an explicit expression for the second term of the asymptotic expansion of large solutions in an annulus. This result shows that the mean curvature of the domain plays a crucial role in the second order terms describing the asymptotic behavior of large solutions. These results were proved under very general conditions on the nonlinearity. The question whether these conditions are optimal remains open. These results have been extended by Bandle and Marcus [12] to general smooth domains. The main result, stated in Theorem 6.3.6, is based on a series of delicate estimates of large solutions near the boundary, expressing the equation in terms of flow coordinates.

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Index

absorption, v fading absorption, 87, 147, 150, 195 Arzela–Ascoli theorem, 118 barrier, viii, 72, 89, 150, 204 global barrier condition, 72, 79, 91 boundary Harnack principle, v, 28, 97, 140 boundary trace, v–vii, 13, 32, 40, 64, 155 rough boundary trace, 74, 81, 94, 107 capacity Bessel capacity, 193 Newtonian capacity, 32, 167 covariant derivative, 126, 141 curvature, 131 Gaussian curvature, 160, 231 mean curvature, 229, 237 principal curvature, 233 Emden-Fowler equations, 113, 159 energy methods, 115, 129, 146, 160 energy function, 126, 135 exhaustion, 13 C ˛ exhaustion, 14 exponential order of growth, 158, 169 Fatou’s lemma, 7, 35, 96 Fatou’s theorem, 173 global orbit, 137 heteroclinic global orbit, 138 Harnack inequality, 76, 121, 160 Hausdorff Hausdorff distance, 181, 189 Hausdorff measure, 14, 172 Herglotz–Doob theorem, 18 Herglotz, vi, 32, 67

Hopf’s principle, lemma, 29, 30, 93, 103, 132 isotropy principle, 122, 129, 156 Kato’s inequality, 20, 32, 71, 114, 130, 139 Keller–Osserman condition, estimate, vii, 71, 88, 108, 130, 162 extended Keller–Osserman condition, 108, 162, 166 global Keller–Osserman condition, ix, 85, 88, 112 local Keller–Osserman condition, 79, 85, 88, 99, 150 positive Keller–Osserman condition, 111, 165, 184, 191, 226 strong local Keller–Osserman condition, 85 kernel averaghing kernel, 20 Green kernel, function, 1, 43, 64, 163, 224 Martin kernel, vii Newtonian kernel, 116 Poisson kernel, ix, 1, 94, 134 Lane–Emden equations, 159 Laplace–Beltrami operator, 132 large solution, vii, 79, 166, 167, 174, 217, 226 asymptotics, x, 159 minimal and maximal large solutions, 184 radial large solution, 175 uniqueness of large solution, ix, 175, 188, 193, 203, 226 limit set, 126, 135, 143, 146, 160

248 Liouville Liouville equation, 160 Liouville type theorem, 113 maximal solution, vii, 72, 78, 85, 89, 159, 162, 192, 196, 211 moderate solution, vii, 66, 78 sigma moderate solution, 193, 236 reflection, 125, 134 regular points (of the boundary trace), viii, 70 regular part of boundary trace, 74, 82 removable boundary singularities, vii, 82, 94, 114, 158 removable sets, 82 Schauder’s estimates, 148 Schauder’s fixed point theorem, 39 semigroup, 137 separable solutions, 114, 120, 147 similarity transformation, 88, 107, 115 singular points (of the boundary trace), viii, 70 singularities (isolated s.), 130, 147, 205 boundary s., vii, 108, 114, 130, 147 interior s., 195 positive boundary s., 116 signed boundary s., 124

Index

strong s. (very singular solution), ix, 96 weak s., 147, 160, 168, 188, 192 Sobolev space, vii, 193 Sobolev imbedding theorem, 11, 143 Sobolev trace, 8, 14, 69 Spectrum, 128 eigenfunction, 133, 156, 210 eigenspace, 129 eigenvalue, 129, 156 Spherical harmonics, 156 Strong maximum principle, 30, 123, 132 Subcritical, vi, 43, 65, 70, 94, 105, 161, 173, 194 Subcriticality condition, vi, 44, 56, 65 Subharmonic, superharmonic, 18, 110 Subsolution, supersolution, 20, 38, 40, 42, 111 Tight (for a sequence of measures), 10 Trajectory, 126, 136, 160 Weak continuity, 44, 52, 56, 65 Weak convergence (in various measure spaces), 9, 10 Weak solution, 2, 4, 10, 27, 33–43, 59, 64, 111, 114, 116 Wiener condition, vii, 32, 167, 187