Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Maximum Principles and Applications, Volume 1

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Yihong Du

Order Structure and Topological Methods in Nonlinear Partial Differential Equations voi 1 Maximum Principles and Applications

World Scientific

Order Structure and Topological Methods in Nonlinear Partial Differential Equations ^|i

Maximum Principles and Applications

Series on Partial Differential Equations and Applications Series Editor: Fang-Hua Lin (Courant Institute of Math. Sci., New York University)

Published Vol. 1

Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics by B. Helffer (Univ. Paris-Sud, France)

Vol. 2

Order Structure and Topological Methods in Nonlinear Partial Differential Equations Vol. 1 - Maximum Principles and Applications by Yihong Du (Univ. of New England, Australia & Qufu Normal Univ, China)

Order Structure and Topological Methods in Nonlinear Partial Differential Equations #>i i

Maximum Principles and Applications

Yihong Du University of New England, Australia & Qufu Normal University, China

\fp

World Scientific

NEW JERSEY • LONDON • SINGAPORE • B E I J I N G • S H A N G H A I • H O N G K O N G • TAIPEI • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Partial Differential Equaitons and Applications — Vol. 2 ORDER STRUCTURE AND TOPOLOGICAL METHODS IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Vol. 1: Maximum Principles and Applications Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-624-4

Printed in Singapore by Mainland Press

Preface

This is volume one of a two volume series. The intention is to provide a reference book for researchers in nonlinear partial differential equations and nonlinear functional analysis, especially for postgraduate students who want to be led to some of the current research topics. It could be used as a textbook for postgraduate students, either in formal classes or in working seminars. In these two volumes, we attempt to use order structure as a thread to introduce the various versions of the maximum principles, the fixed point index theory, and the relevant part of critical point theory and Conley index theory. The emphasize is on their applications, and we try to demonstrate the usefulness of these tools by choosing applications to problems in partial differential equations that are of considerable concern of current research. An important work in this direction is H. Amann's classical review article (SIAM Rev. 18 (1976), 620-709), which discussed the combination of order structure and fixed point index theory and its applications to various problems of nonlinear partial differential equations. Much progress has been made since this article. The fixed point index theory has been further developed and found important new applications in partial differential equations. Moreover, the order structure has since been successfully combined with critical point theory and Conley index theory to study various nonlinear partial differential equation problems. Furthermore, the classical maximum principle in partial differential equations has found new applications in several important problems. All these are scattered in research articles published in various professional journals, and most of them are still active topics of current research. It is our hope that through these two volumes, we can present the reader

v

VI

Maximum

Principles

and

Applications

in a somewhat systematic way some of the new progresses in these topics. As the title suggests, volume 1 mainly considers the maximum principles and their various applications in some of the current research topics. The topological methods will be discussed in volume 2. There are 7 chapters and an appendix in this volume 1. In chapter 1, we use the Krein-Rutman theorem to derive several well-known properties of the principal eigenvalues, we then use these in chapter 2 to characterize the maximum principle. We briefly discuss the moving plane method in chapter 3. Existence results are not discussed until chapter 4, where we consider the methods of upper and lower solutions, also known as super and sub-solution methods. The weak theory here is based on the theory of monotone operators, whose basic result is recalled without proof. With these preparations, existence results can be considered in the later chapters. In chapter 5, the basic logistic model is discussed, where various comparison arguments, the upper and lower solution methods, together with a variety of elliptic estimates are used. Chapter 6 gives an introduction of some basic boundary blow-up problems. The last chapter considers again various symmetry properties of elliptic problems, where apart from the moving plane methods, other techniques are also used. In the appendix, we include a brief review of the classical elliptic theory for second order partial differential equations. Since this basic theory may take a long time for the beginners to master, we feel it might be practical for those readers such as postgraduate students to initially accept the relevant basic results in this theory and continue with their study of some current research topics. Some of the material is chosen to be included here for its usefulness, such as the various versions of the maximum principles, and the upper and lower solution methods. In such a case, we have tried to make the results as general as possible, provided that not too much complication of the presentation is caused. Some of the material is included here in order to introduce useful techniques and to lead the reader to some of the current research problems. In this situation, we usually put clarity in front of generality. The material presented and the references quoted here are mainly based on the author's taste and familiarity, which inevitably are biased with many important topics and references not included here. I apologize if these omissions inadvertently offend anyone. In volume 2, we will discuss some developments of the fixed point index theory (mainly due to E.N. Dancer), and their applications to various problems, in particular to several population models. We will also discuss the

Preface

VII

part of critical point theory and Conley index theory that can be combined with order structure to provide better applications. Some of the material in volume 1 here provides necessary preparation for volume 2. It is my great pleasure to thank all those who helped in one way or another in the writing of this first volume. In particular, I would like to express my deep thanks to Professor Norman Dancer for guiding me into nonlinear analysis, and for the constant help and encouragements. My sincere thanks to Professor Dajun Guo for taking me into nonlinear functional analysis, and to Professor Xingbin Pan for encouraging me to write this book. I'm grateful to my colleagues at the University of New England who freed me from teaching duties in the first half of 2005; that helped immensely in getting this belated volume ready before the end of the year. Part of the material here was presented at Qufu Normal University at a workshop in 2004, and my thanks go to the colleagues there for the help and support. Over the years, I have benefitted greatly from working with my collaborators. My sincere thanks to all of them, in particular, Florica Cirstea, Zongming Guo, Shujie Li, Lishan Liu, Li Ma, Tiancheng Ouyang, Shusen Yan, Feng Zhou, and my former PhD students and friends Qingguang Huang and Wei Dong, whose joint papers with me are used in this volume. Several friends helped me proof reading various parts of this volume, and I would like to thank in particular Florica Cirsta, Xing Liang, Rui Peng and Shusen Yan for their efforts in helping me reducing the mistakes. It is my responsibility for any remaining mistakes, and corrections from the readers are very much appreciated. My sincere thanks also go to the editors of World Scientific Publishing, especially Ms Zhang Ji, for all the help and advices. Finally I thank my family for the understanding and support during the writing of this book.

Yihong Du September, 2005, Armidale

This page is intentionally left blank

Contents

Preface

v

1.

Krein-Rutman Theorem and the Principal Eigenvalue

1

2.

Maximum Principles Revisited

9

3.

4.

5.

6.

2.1 Equivalent forms of the maximum principle 2.2 Maximum principle in W2'N(Cl)

9 11

The Moving Plane Method

17

3.1 Symmetry over bounded domains 3.2 Symmetry over the entire space 3.3 Positivity of nonnegative solutions

17 23 28

The Method of Upper and Lower Solutions

33

4.1 Classical upper and lower solutions 4.2 Weak upper and lower solutions

33 39

The Logistic Equation

61

5.1 The classical case 5.2 The degenerate logistic equation 5.3 Perturbation and profile of solutions

61 64 75

Boundary Blow-Up Problems

83

6.1 The Keller-Osserman result and its generalizations 6.2 Blow-up rate and uniqueness 6.3 Logistic type equations with weights ix

84 95 102

X

7.

Maximum

Principles

and

Applications

Symmetry and Liouville Type Results over Half and Entire Spaces

117

7.1 7.2 7.3 7.4

117 128 139 145

Symmetry in a half space without strong maximum principle Uniqueness results of logistic type equations over RN . . . . Partial symmetry in the entire space Some Liouville type results

Appendix A A.l A.2 A.3 A.4 A.5

Basic Theory of Elliptic Equations

Schauder theory for elliptic equations Sobolev spaces Weak solutions of elliptic equations LP theory of elliptic equations Maximum principles for elliptic equations A.5.1 The classical maximum principles A.5.2 Maximum principles and Harnack inequality for weak solutions A.5.3 Maximum principles and Harnack inequality for strong solutions

163 163 166 170 174 177 177 178 179

Bibliography

181

Index

189

Chapter 1

Krein-Rutman Theorem and the Principal Eigenvalue The Krein-Rutman theorem plays a very important role in nonlinear partial differential equations, as it provides the abstract basis for the proof of the existence of various principal eigenvalues, which in turn are crucial in bifurcation theory, in topological degree calculations, and in stability analysis of solutions to elliptic equations as steady-state of the corresponding parabolic equations. In this chapter, we first recall the well-known KreinRutman theorem and then combine it with the classical maximum principle of elliptic operators to prove the existence of principle eigenvalues for such operators. Let X be a Banach space. By a cone K C X we mean a closed convex set such that XK C K for all A > 0 and K n (-K) = {0}. A cone K in X induces a partial ordering < by the rule: u < v if and only if v — u G K. A Banach space with such an ordering is usually called a partially ordered Banach space and the cone generating the partial ordering is called the positive cone of the space. If K — K = X, i.e., the set {u — v : u, v £ K} is dense in X, then K is called a total cone. If K — K = X, K is called a reproducing cone. If a cone has nonempty interior K°, then it is called a solid cone. Any solid cone has the property that K — K = X; in particular, it is total. Indeed, choose XQ G K° and r > 0 such that the closed ball Br(uo) := {u G X : \\u — UQ\\ < r} is contained in K. Then for any u G X\{0}, v0 := uo+ru/||u]| e K and hence u = (||u||/r)(v 0 —«o) G K—K. We write u>viiu~v£K\ {0}, and u > w i f w - « € K°. Let X* denote the dual space of X. The set K* := {I G X* : l{x) > 0 Vx G K] is called the dual cone of K. It is easily seen that K* is closed and convex, and XK* C K* for any A > 0. However it is not generally true that K* n (-K*) = {0}. But if K is total, this last condition is satisfied and hence K* is a cone in X*. Indeed, if / G K* n (-K*), then for every

1

2

Maximum

Principles

and

Applications

x £ K, l(x) > 0, — l(x) > 0, and therefore l{x) = 0 for all x £ K. Since K — K = X, this implies that l(x) = 0 for all i £ l , i.e., / = 0. Let Q be a bounded domain in RN. It is easily seen that the set of nonnegative functions K in X = LP(Q.) is a cone satisfying K — K = X. However, it has empty interior. Similarly the set of nonnegative functions in W1'P{Q) gives a reproducing cone, and generally the nonnegative functions in Wk^{D) (k > 2 ,p > 1) form a total cone. On the other hand, the nonnegative functions form a solid cone in C(fl) but only form a reproducing cone in Co(O) := {u e C(£l) : u = 0 on 9Q}. If fi has C 1 boundary dQ, then it is easy to see that the nonnegative functions in Co(fi) := {u 6 C 1 (fl) : u = 0 on dQ} form a solid cone; for example, any function satisfying u(x) > 0 in Q and Dvu{x) < 0 on dQ is in the interior of the cone, where v denotes the outward unit normal of dfi. T h e o r e m 1.1 (The Krein-Rutman Theorem, [Deimling(1985)] Theorem 19.2 and Ex.12) Let X be a Banach space, K C X a total cone and T : X —* X a compact linear operator that is positive (i.e., T{K) C K) with positive spectral radius r(T). Then r(T) is an eigenvalue with an eigenvector u £ K \ {0}; Tu = r(T)u. Moreover, r(T*) = r(T) is an eigenvalue ofT* with an eigenvector u* G K*. Let us now use Theorem 1.1 to derive the following useful result. Theorem 1.2 Let X be a Banach space, K C X a solid cone, T : X —* X a compact linear operator which is strongly positive, i.e., Tu ^> 0 if u > 0. Then (a) r(T) > 0, and r(T) is a simple eigenvalue with an eigenvector v £ K°; there is no other eigenvalue with a positive eigenvector. (b) |A| < r(T) for all eigenvalues A ^ r(T). Let us recall that r is a simple eigenvalue of T if there exists v ^ 0 such that Tv = rv and (rl — T)nw — 0 for some n > 1 implies w e span{v}. Proof. Step 1: There exists v0 > 0 such that Tv0 = r(T)v0. Fix u £ K°. Then aTu > u for some a > 0, and we can find a > 0 such that Btj{u) c K. It follows that w < (cr) _1 ||ui||u for any w £ X. Let S = aT. Then u < Snu < a-^l^uWu

< <7- 1 ||S n ||||u||i(, Vn > 1.

Krein-Rutman

Theorem and the Principal

Eigenvalue

3

Hence \\Sn\\ > a/\\u\\ and r{S) = lim \\Sn\\^n > 0. By Theorem 1.1, r(S) is an eigenvalue of S corresponding to a positive eigenvector v0 £ K \ {0}. Clearly r(T) = r(S)/a > 0 and Tv0 = r(T)v0. Step 2: To prove that r(T) is simple, we show a more general conclusion: lfr>0 and Tv = rv for some v > 0, then r is a simple eigenvalue

ofT. Let us first show that (rl — T)w = 0 implies w £ span{v}. Suppose Tw = rw with w ^ 0. Then T(v ± tw) = r(v ± tw) for all t > 0. Since T is strongly positive, v £ K° and the above identity implies v ± tw $ dK unless v ± tw = 0. But v±tw £ K° for small t and this cannot hold for all large t for otherwise w £ K D (—-ft") = {0}. Therefore there exists t0 ^ 0 such that u -f t0w £ dK and hence v + tow — 0. This proves w £ span{v}. Let (rl — T)2w = 0. By what has just been proved, rw — Tw = tov for some t0 £ R1. If t 0 ^ 0, then we may assume to > 0 (otherwise change w to — iu). Since T(t; + sw) = r(v + sw) — sj^of 0, and v + sw £ K° for all small s > 0, we easily deduce v + sw £ K° for all s > 0. This implies that w £ K, and hence w = r _ 1 ( i o ^ + Tw) £ K°. We now have w -tv

£ K° for all small i > 0,

but not for all large t > 0 as this would imply v — 0. Therefore there exists t\ > 0 such that w — t\v £ dK. But then rw — tov — £irt> = T(w — *iv) > 0, w — t\v > r~ltoV 3> 0, contradicting w — t\v £ dK. Therefore we must have to = 0 and hence rw — Tw = 0, w £ span{v}. This proves that r is a simple eigenvalue. Step 3: Next we show that T cannot have two positive eigenvalues r\ > T2 corresponding to positive eigenvectors: Tv\ —r\V\, Tv2 — r2V2Let v(t) = V2 — tv\, t > 0. Since T is strongly positive, we have v\, v2 £ K°. As before we have v(t) £ K° for small t but not for all large t.

4

Maximum

Principles

and

Applications

Therefore there exists to > 0 such that v(to) £ K but v(t) g K for t > to. We now have v2 - to(n/r2)vi

= r^Tfa

- t0Vi) £ K,

which implies r\ < r2 due to the maximality of to- This contradiction proves step 3. S t e p 4: IfTw = Aw with w =£ 0 and A ^ r{T), then |A| < r(T). If A > 0, then by Step 3, w £ K. It follows that v0 + tw £ K for all small i > 0 but not for all large t. Therefore there exists to > 0 such that vo + tow £ K and vo + tw ^ K for t > to- It then follows that v0 + t0(X/r(T))w = r ( r ) ~ x T ( v 0 + t0w) £ K. The maximality of t0 implies that A < r(T) and hence A < r(T). If A < 0, then from T2w = \2w and T2v0 = r(T)2v0 and the above argument (applied to T 2 ) we deduce A2 < t ( T ) 2 and hence |A| < r(T). Consider now the case that A = o + ir with T / 0 . Then necessarily w = u + iv and Tu = ou — rv, Tv =

TU +

av.

(1-1)

We observe that u and v are linearly independent for otherwise we necessarily have r = 0. Let X\ := span{u,v}. Then (1.1) implies that Xi is an invariant subspace of T. We claim that K\ :— X\C\K — {0}. Otherwise K\ is a positive cone in X\ with nonempty interior, as for any w £ K\\ {0}, Tw £ Xi n K° - K^. We can now apply Step 1 above to T on Xi to conclude that there exists r > 0 and WQ £ K® such that Two = rwo- By Steps 2 and 3, we necessarily have r = r(T) and Wo € span{vo). In other words, t;o £ K\ and «o = &u + flv for some real numbers a and /3. But then one can use (1.1) and TVQ = r(T)vo to easily derive a = j3 = 0, a contradiction. Therefore K\ = {0}. From span{u, v}C\K = {0} we find that the set S := {(£, ri)£R2:v0+tu

+ riv£

K}

is bounded and closed. Since v0 £ K°, M := sup{£ 2 + rj2 : (£,r?) £ £ } > 0 and is achieved at some (£o,?7o) G S. Let 2 0 = VQ -\- ^Qu + r]ov. Then 2 0 £ K \ {0} and Tz0 G X°. Therefore we can find a £ (0,r(T)) such that T^o > oivo, i.e., ( r ( T ) - a ) v 0 + ( 6 « + »?iu)>0,

(1.2)

Krein-Rutman

Theorem and the Principal

Eigenvalue

5

where £1 = £o<7 + VOT, 771 = rjocr - £0T.

Clearly

By (1.2), we find that ( 6 , TJI )/(r(T) -a)

ell and hence

£2+772<M(r(T)-a)2, that is,

\\\2<(r(T)-af, and hence |A| < r(T). The proof of Step 4 and hence the theorem is now complete. • Suppose now L is the elliptic operator and ft the bounded domain as given in Theorem A.4, namely Lu = alJ(x)DijU + b%(x)DiU + c(x)u has C a (Q) coefficients and is strictly uniformly elliptic in the bounded domain ft which has C2'a boundary. Choose £ > 0 large enough so that c — £ < 0 in ft, and denote L^u = Lu — £u. Let K be the positive cone in X := C 0 ' a (ft) consisting of nonnegative functions. For any v e X, Theorem A.l guarantees that the problem —L^u = v in ft, u = 0 on <9ft has a unique solution u satisfying

IMkaa for some constant C\ > 0 independent of u and v. It follows that T : X —> X defined by Tv = u is a compact linear operator. Moreover, by the weak maximum principle Theorem A.34, Tv > 0 if v € K. The strong maximum principle Theorem A.36 then implies that u = Tv > 0 in ft if v € K \ {0}, and the Hopf boundary lemma (Lemma A.35) gives further Dvu < 0 on 9ft. This implies that Tv € K°. Therefore T is strongly positive. It now follows from Theorem 1.2 that r(T) > 0 is a simple eigenvalue of T with an eigenfunetion v € K°: Tv = r(T)v. Thus u = Tv satisfies —Lu + £u = r(T)~1u

in ft, u = 0 on 9ft,

Maximum

6

Principles and

Applications

i.e., Lu + \\u

= 0 in ft, u = 0 on dCl,

with Ai = r ( T ) - 1 - £ . Generally, it is easily checked t h a t [i is an eigenvalue of T if and only if A = n~l — £ is an eigenvalue of Lu + AM = 0 in ft, u = 0 on 5ft.

(1.3)

Theorem 1.2 now implies the following result. T h e o r e m 1.3 Under the conditions of Theorem A.4 for L and ft, the eigenvalue problem (1.3) has a simple eigenvalue Ai £ R1 which corresponds to a positive eigenfunction; none of the other eigenvalues corresponds to a positive eigenfunction. If the boundary operator is of Neumann or Robin type, Bu = Dvu + a(x)u,

a > 0, a £ C 1 , a ( d f i ) ,

then we let X = C 1 , Q (ft) and let K be the cone of nonnegative functions in this space. We define the operator T analogously as in the Dirichlet case and again find t h a t it is compact on X and maps K t o itself, due to the weak maximum principle. Suppose now v £ K \ {0}. T h e n by t h e strong m a x i m u m principle, u = Tv > 0 in ft. Moreover, by t h e Hopf b o u n d a r y lemma, if U(XQ) = 0 for some XQ € dCl, then D^u^o) < 0 and hence BU(XQ) < 0, contradicting the boundary condition. Therefore u > 0 on dCl. Therefore Tv > 0 on ft, which implies t h a t Tv £ K°, i.e., T is strongly positive. Therefore we can apply Theorem 1.2 t o conclude t h a t Theorem 1.3 holds also for the Neumann and Robin b o u n d a r y conditions. T h e eigenvalue Ai in Theorem 1.3 is usually called the principle eigenvalue. T h e o r e m 1.4 If X ^ Ai is an eigenvalue of (1.3) but the boundary condition is either Dirichlet, or Neumann, or Robin type, then Re(X) > X\. Proof. Suppose w > 0 is an eigenvector corresponding to Ai and u is an eigenvector corresponding to A. Set v := u/w. Then —Xv = w~1L(vw)

= Lv — cv + 2w~lali

DjwDiV

— Xiv.

Writing Kv := aijDijV

+ VDiV, V := 6 i +

2vj-1aijDjw,

Krein-Rutman

Theorem and the Principal

Eigenvalue

7

we obtain Kv + (X - \i)v = 0. Take complex conjugates to yield Kv + (X-Xi)v

= 0.

Next we compute K(\v\2) = K(vv) = vKv + vKv + 2aijDivDjV

> vKv + vKv,

since aij^j

= a^(Re{^)Re(^)

+ Im^Imfa))

>0

for any complex vector £ £ CN. We now easily obtain i f ( H 2 ) > 2(i?e(A) - Ai)M 2 in ft. Suppose now the boundary operator B is either Neumann or Robin type. Then w > 0 over ft and a direct computation shows Dvv = 0 and D „ H 2 = 0. If Re(X) < Xi, then 0 := \v\2 > 0 satisfies K4> > 0 in ft, Dv(j> = 0 on 9ft. We now apply the strong maximum principle and Hopf boundary lemma and conclude that <> / = constant, that is u = cw and hence A = X\, a contradiction. Therefore we must have Re(X) > Ai. To prove the Dirichlet case, we replace w by ive := w1_e, 0 < e < 1, in the above discussion and obtain K{\v\2) > -2(Re(X) + — )\v\2 in ft. we Since Lwe = (1 - e)w~eLw - e(l - e)u> - 1 - V J DiivDjW + ecu)1-* < (1 - e)w~eLw + ecw 1 - 6 < (eC - (1 - e)Ai)iue, where C = max^c, we deduce K(\v\2) > 2((1 - e)A! - eC - Re(X))\v\2 in ft.

8

Maximum Principles and Applications

By the Hopf boundary lemma, we know Duw < 0 on d£l. Therefore, since u\gn = 0, u(x) _ z-txoGdfi w(x)

Duu(xp) DVW{XQ)

It follows that ii(x\

lim v — lim w(x)e—r-r x-*dQ

x-tdn

= 0.

W(X)

If Re(X) < (1 - e)Ai - eC, then, K(\v\2)>0ma. Let Qn be a sequence of smooth domains enlarging to Q, e.g., Qn := {a; £ fi : d(a;, 3Q) > So/n} with <5o > 0 small. We apply the maximum principle on Cln and deduce max^ |t>|2 < maxan^ \v\2. Letting n —• oo, it results \v\2 = 0 and hence u = 0, a contradiction. Therefore Re(\) > (1 — e)Ai — eC. Letting e - > 0 w e obtain Re(X) > \i• Remark 1.5 The above proof shows that in the Neumann and Robin boundary conditions case, Re(\) > \\. This is also true in the Dirichlet case; see Theorem 2.7 in the next chapter. Remark 1.6 Instead of (1.3), sometimes one also needs to consider the weighted eigenvalue problem Lu + \h(x)u = 0 in Q, Bu = 0 on d£l, where h(x) is a weight function, B is the Dirichlet, Neumann or Robin boundary operator. If h(x) is positive and suitably smooth, similar results to those in Theorems 1.3 and 1.4 can be analogously proved. If h(x) changes sign, similar results can still be proved by considerably different techniques, see [Hess-Kato(1980)].

Chapter 2

Maximum Principles Revisited

In this chapter we make use of the principal eigenvalues to formulate a useful characterization of the so called maximum principle property. The first section considers the classical case for C 2 solutions and the second section discusses solutions in W2'N.

2.1

Equivalent forms of the maximum principle

The principal eigenvalue can be used to formulate a necessary and sufficient condition for the validity of the maximum principle. In order to state the results in a concise way, we use a well-known framework that incorporates all three boundary conditions considered in the last chapter and also the mixed boundary conditions. We assume that To and Ti are two disjoint open and closed subsets of dVt with TQ U T\ = dfl, and define Bu:=l"

on To, [ Uvu + au on 11,

where a 6 C 1 , a (Q) is nonnegative. It is possible that either To or I \ is the empty set. As in the previous chapter, we assume that Q is bounded with C 2 , Q boundary and L is as in Theorem A.4. It is easy to modify the proofs in the previous chapter to obtain the following result. Theorem 2.1

The eigenvalue problem Lu + Xu — 0 in n, Bu = 0 on dfl

has a principal eigenvalue Ai = X\(L,B, ft), and any other eigenvalue A satisfies Re(X) > Ai. 9

10

Maximum

Principles and

Applications

Definition 2.2 We say that (L, B, Cl) has the strong maximum principle property if u G C2(Cl) n C1 (Cl U Ti) n C(Cl U To) and - L u > 0 in Cl, Bu > 0 on dCl imply u > 0 in Cl unless u = 0. Definition 2.3 of (L, B, Cl) if

A function u £ C2(Cl) n C 1 (fi) is called a supersolution - L u > 0 in Cl, Bu > 0 on df2.

It is called a strict supersolution if it is a supersolution but not a solution. Theorem 2.4 Under the above assumptions on (L,B,Cl), statements are equivalent:

the following

(i) (L,B,Cl) has the strong maximum principle property; (ii) (L,B,Cl) has a strict supersolution which is positive in CI; (Hi) Xi(L,B,Cl) > 0 . Proof. (i)=^(iii). Arguing indirectly, we assume Ai < 0. We may choose the corresponding eigenfunction of Ai negative: 4> < 0 in Cl. Therefore -L(p = Ai<£ > 0 in CI, B<j> = 0 on dCl.

By the strong maximum principle this implies 4> > 0, a contradiction. (iii)=^(ii). Let tp > 0 be the corresponding eigenfunction of Ai. Then - L V = AiV> > 0 in CI, Bij> = 0 on dCl. Hence I/J is a strict supersolution that is positive in CI. (ii)=^(i). Let v be a strict supersolution, positive in Cl. We first claim that v > 0 on Ti. Otherwise, that exists XQ € Ti such that v(x0) = 0. By the Hopf lemma, we deduce Bv(x0) = Duv(x0) < 0, contradicting the assumption that Bv > 0 on I V Next we show that v(x) > Sd(x,To) for some 5 > 0 and all x £ Cl. Otherwise we can find a sequence xn 6 Cl such that v(xn) < (l/n)d(xn, To). In particular, v(xn) —-> 0 as n —» oo. As v is positive in fiuri, we necessarily have, by passing to a subsequence if needed, xn —» XQ £ To. It follows by the continuity of v that V(XQ) — 0, and hence DVV(XQ) < 0 by the Hopf lemma. But this last inequality implies v(x) > Sd(x,T0) for some 6 > 0 and all x E Cl n BT(XQ), where BT(XQ) denotes a small ball centered at x$; this is in contradiction to the inequalities for xn with large n.

Maximum

Principles

Revisited

11

We are now ready to show that (L, B, ft) has the strong maximum principle property. Suppose that u ^ 0 and satisfies -Lu > 0 in ft, Bu > 0 on 5ft. If u > 0 in ft then there exists XQ £ ft such that U(XQ) > 0, as we have assumed u ^ 0. If there exists x\ £ ft such that u(x{) = 0, then we can find an open ball B C ft such that u{x) > 0 in 5 but u(x2) = 0 for some X2 € <9.B n ft. We can now apply the Hopf lemma at x2 to derive that D„0u(x2) < 0, where VQ denotes the unit outward normal of dB at x2. But this implies that u(x) < 0 for some x £ ft \ B close to X2, a contradiction. Therefore we must have u > 0 in ft. The only other possibility is that min^u = u(xo) < 0. Since u > 0 on To and u € C 1 (ft), we easily see that for some M > 0 large, u > —Md(x, To). Therefore, recalling v > Sd(x,ro), we have u + £v > 0 in ft, V£ > M/8. This does not hold for small £ > 0 as U{XQ) < 0. Therefore, there exists a smallest positive £, say £o, such that u + £QV > 0 in ft. Now v := u + £0v is a strict supersolution that is nonnegative in ft. It is not identically zero because this would force v to be a solution due to the inequality satisfied by u. Therefore the argument in the previous paragraph can be repeated to show v > 0 in ft. Then, as for v before, we can find 8 > 0 such that v > 8d(x, To) and hence u + £o^ > —{8/M)u in ft, that is u + £o(l + 8/My1v

> 0 in ft,

contradicting the minimality of £o- Therefore the case min^u < 0 does not occur. This finishes the proof. • Remark 2.5 The careful reader might have observed that in the above proof we have not used a > 0 in the boundary condition. Indeed, this restriction can be removed; it is also not needed for Theorem 2.1 to hold. We refer to [Amann(1983)] and [Amann-L6pez(1998)] for more details.

2.2

Maximum principle in

W2'N(fl)

Let L be an elliptic operator in a bounded domain ft C RN, of the form Lu = alj(x)Dij

+ bl{x)DiU + c(x)u,

Maximum

12

Principles and

Applications

where a lJ € C(ft), bl,c £ L°°(ft), and for some positive constants CQ and Co, colCI2 < aij(x)Uj

< Col^l2, V£ £ fl", Vx 6 ft.

We say that the maximum principle holds for L and ft if for any v £

w?0f(n), Lv > 0 in ft, Irmx_+,9nv(z) < 0

(2.1)

implies i> < 0 in ft. A useful sufficient condition for the maximum principle of L in ft to hold is the following. Theorem 2.6 Let L be as above. Assume diam(ft) < d. Then there exists 6 > 0 depending on L and d such that the maximum principle for L in ft holds if the measure of ft satisfies |ft| < 5. Proof.

Write c = c+ — c~ and rewrite (2.1) as aijDijV + VDiV - c~v >

-c+v+.

We then apply the Aleksandrov weak maximum principle Theorem A.40 to obtain supv < C\\c+v+\\LN(n)

< Ci maxv+|ft| 1 / J v .

It follows that v < 0 in ft provided that C i j f t l 1 ^ < 1. Therefore we can take 6 = C^N. • In [Berestycki-Nirenberg-Varadhan(1994)], the results in Theorem 2.1 and Theorem 2.4 with Dirichlet boundary conditions are established for solutions in Wl(Jc (ft) with L as above; they do not require any smoothness condition on <9ft. For simplicity of presentation, however, we only prove these results under the extra restriction that 9ft is Lipschitz continuous. (In fact, we only need that ft satisfies an exterior cone condition at every boundary point.) Theorem 2.7 (Principal eigenvalue) Let the above assumptions on L be satisfied and suppose that ft has Lipschitz boundary. Then the following holds:

Maximum

Principles

Revisited

(i) There exists a real number Ai and <j>i G W^(i}) satisfying

13

D C(fi), Vp > N,

(L + Ai)(/>i = 0, <> / > 0 m Q, i = 0 on dfi. (MJ Suppose ip G W,20*(n) n C(H) satisfies (L + X)ip = 0, I/J > 0 in fi, ip — 0 on dCl. Then A = Ai and ^ e's a constant multiple of 4>\. (Hi) Suppose ip G Wj0£(fl) n C(fi) (possibly complex valued) satisfies (L + A)V> = 0 m f], V = 0 on <9ft and A ^ Ax. ITien Re(X) > A1; and i / ^ is real then it changes sign in fi. Proof, (i) We will use the Lp theory for elliptic operators, the strong maximum principle and the Krein-Rutman theorem. Choose £ > 0 so that c — ^ < 0 in fi, and denote L^u = Lu — £u. Let K be the positive cone in X :— Co(fl) consisting of nonnegative functions. For any v G C(Q) c Lp{Vt), Theorem A.33 guarantees that the problem —L$u = v in fi, w = 0 on dQ, has a unique solution u G W ^ ( f i ) n C(ft), Vp > 1. Clearly T : X -> X defined by Tv = u is a linear operator. Let us now show that it is bounded and compact. Let u\ = T(||u||oo), where |)u||oo = sup n v. We find that —Lj(u\ — u) > 0 in fi, («i — u) = 0 on 9 0 . Applying the Alksandrov weak maximum principle (Theorem A.40) we deduce u\ — u > 0 in fi, i.e., Tv < T(||u||oo) = IMIooT'fo, uo = 1 in &• Similarly, Tv > - H ^ T u o - Therefore, ||Tu||oo < M||t>||oo, M = | | T M 0 | U . So T is bounded. To see that T is compact, let vn be a bounded sequence in X, say ||wn||oo < C- Then from VQ := TUQ —> 0 uniformly as a; —> <9S7 and — Ct>o < r u n < CVQ in Q, we see that Tvn —> 0 uniformly in n and —> dfi. For any fi' CC fi, by the interior estimate (Theorem A.26) and the Sobolev imbedding theorem, we find that {Tvn} is a bounded set in C 1 (fi'). Thus {Tvn} is bounded and equicontinuous in C(Cl). Hence it is precompact in X, by the well-known Arzela-Ascoli theorem. This proves that T is compact. Moreover, by the strong maximum principle (Theorem A.41), if v G K \ {0}, then Tv > 0 in ft. This fact implies that T is irreducible (see [Sweers(1992)] page 254 for a proof) and hence r(T) > 0 by [de Pagter(1986)]. As clearly K is reproducing, we can apply the Krein-Rutman

14

Maximum

Principles and

Applications

theorem to conclude that there exists v G K \ {0} such that Tv = r(T)v. Therefore (pi :=Tv satisfies (pi > 0 in ft and Lcfii + Ai^i = 0 in Q,, (pi = 0 on <9Q, with Ai = £ — r{T)_1. Applying Theorem A.33 to the above identities we find 4>! G Wfo'cp(0) n C(ft), Vp > N. This proves conclusion (i). (ii) We make use of Theorem 2.6 above. Firstly we choose a smooth subdomain K CC f2 such that jfi \ K\ is small enough such that the maximum principle for £(-A) in Q' :— f2 \ K holds. We can now find t > s > 0 such that sip < (pi < tip in if. For definiteness, we assume that A < Ai; the case A > Ai can be treated similarly. Therefore L{4>\ - sip) = Xstp - Ai^i < -\( 0 on dCl'. By Theorem 2.6, we deduce 0 in 0'. Therefore 4>i — sip > 0 in Q. Let so be the largest s satisfying the above inequality in Q. Then we have 4>\ — SQip > 0 in Q, and for all large £, L(i — SQip = 0 or (pi — s0ip > 0 in fl. If the first alternative occurs, then we necessarily have A = Ax and tp is a constant multiple of 4>i- If the second alternative occurs, then we can find s\ > s0 such that s\ip in K and then apply Theorem 2.6 to 0 in fi, contradicting the maximality of so- Therefore, ip must be a constant multiple of (p\ and A = Ai. This proves conclusion (ii). (iii) If ip is real (and hence A must be real), then by (ii) ip must change sign in Q. If A < Ai then by considering the maximal so > 0 such that 0 in Q as in the proof of (ii) above, we deduce a contradiction in a similar way. Therefore A > Ai in this case. Suppose now ip is a complex function. Consider the product domain ft = fl x fl, with points (a;, y), x G Q, y G Q. Let Lx denote the action of L in the x variables, and Ly its action in the y variable and set L — Lx + Lv. It is easily checked that Lfa + 2Ai^i = 0 in Q,, (pi = 0 on d(l,

Maximum

Principles

Revisited

15

where <j>i(x,y) := <j>i(x)4>i(y) > 0. Therefore by conclusions (i) and (ii) applied to (L, ft) we find that 2X\ is its principal eigenvalue. (We note that ft has Lipschitz boundary.) By a direct computation, tp(x,y) := ip(x)tp(y) + / ip(x)tp(y) is a real valued function satisfying Lip + (A + X)tp = 0 in ft, -0 = 0 on dft. We now apply our argument in the last paragraph and conclude that A+A > 2Ai, i.e., Re(X) > Ai. This finishes the proof of conclusion (hi). • Using the technique in the proof of (ii) above, we can now prove the following result by arguing similarly as in the proof of Theorem 2.4. The details are left to the reader as an exercise. Theorem 2.8 Let the above assumptions on L hold and suppose that ft has Lipschitz boundary. Then the following are equivalent: (i) The maximum principle holds for L in fl. (ii) There exists (/> £ Wf^ (fl) n C(Tl) such that L(f> < 0 in ft, 4> > 0 on 9ft, and at least one inequality cannot be replaced by equality. (Hi) The principal eigenvalue \\ > 0. Remark 2.9 If 5ft is smooth enough, say C 2 , then we can apply the global Lp theory and prove similar results to Theorems 2.7 and 2.8 for Neumann and Robin (or mixed) boundary value problems. Similar results can also be proved for systems of equations that are cooperative in nature; see [Sweers(1992)] for more details. If u G C 2 (ft) has a local maximum at XQ € ft, then clearly DU(XQ) = 0 and [Diju(xo)] is a seminegative definite matrix. Let u e W^0'c (ft). Then by Theorem A.6, the classical partial derivatives Diu(x) and Dijii(x) exist for a.e. x in any ft' CC ft. The following theorem extends the above mentioned result for C 2 functions to W^c (ft) functions to some extent, and is known as Bony's maximum principle. (Bony first proved the result for W^'P(ft) functions with p > N, see [Bony(1967)]; P.L. Lions then extended it to include the case p = TV.) Theorem 2.10 ([Lions(1983)] Corrolary 2) Suppose u £ W^(fl) has a local maximum at XQ £ ft. Then there exists a subset A of fl such that the

16

Maximum

Principles

and

Applications

classical derivatives Dtu and DijU exist on A and \A n S r (a;o)| > 0 for any r > 0; \imx^XO:XeA^o,13Diju(x)

< 0,

lim

|Du(x)| = 0.

x—*xo,x£A

Using Theorem 2.10, it is easy to see that the Hopf boundary lemma (Lemma A.35) still holds for the case where the solution u 6 W2'N(Q). The proof is the same as the standard one. Theorem 2.11 The conclusions in Lemma A.35 still hold when u £ C 2 (n) is replaced by u £ W2>N({1).

Chapter 3

The Moving Plane Method

The moving plane method is a clever way of using the maximum principle to obtain qualitative properties of positive solutions of some elliptic equations, notably the symmetry of such solutions. It was introduced by A.D. Alexandroff in his study of surfaces of constant mean curvature, and successfully used by J. Serrin [Serrin(1971)] in proving symmetry properties for some over-determined elliptic problems. It has become well-known through the works of Gidas-Ni-Nirenberg [Gidas-Ni-Nirenberg(1979)] and [Gidas-Ni-Nirenberg(1981)], where it was used to prove symmetry results for positive solutions of rather general nonlinear elliptic problems. Since then, this method has been further developed and used in a variety of problems by many people. In this chapter, we will look at several symmetry problems and related questions.

3.1

Symmetry over bounded domains

Consider the following elliptic boundary value problem - A u = f(u) in fl, u = 0 on dSl,

(3.1)

where fi is a bounded domain in RN and / is a Lipschitz continuous function. The Laplacian operator Au(x) = T,?=1uXiXi(x) is invariant under many operations on the variable x € RN. For example, it is invariant under translations and rotations. Therefore, if u is a solution to (3.1) and if £1 is invariant under such an operation, say T, then UT{X) :— 17

18

Maximum

Principles

and

Applications

u(Tx) is also a solution. If there is a unique solution, then necessarily UT = u. An important problem it to understand when we always have UT — u even if uniqueness does not hold. In [Gidas-Ni-Nirenberg(1979)], Gidas, Ni and Nirenberg proved, by making use of the maximum principle, that any positive solution of (3.1) is radially symmetric when ft is a ball. The technique they employ is called the "moving plane method". Their original proof was later considerably simplified (see [Berestycki-Nirenberg(1991)]) by making use of Theorem 2.6 in the previous chapter. In this section, we will prove this symmetry result and look at some related problems. Definition 3.1 A domain ft C RN is called Steiner-symmetric with respect to the plane x\ = 0, if it is convex and symmetric in the xi-direction, i.e., for every (xi,x') := (x\,X2, ...,xpj) S ft, {{t,x') : \t\ < |xi|} C ft. Clearly an arbitrary ball in RN is Steiner-symmetric with respect to any plane passing through its center. However, the so called "Star of David" (see Fig. 1) in R2 is Steiner-symmetric with respect to the plane X\ = 0, but not with respect to x-i = 0. Theorem 3.2 Let ft be an arbitrary bounded domain in RN which is Steiner symmetric with respect to the plane x\ = 0. Let u be a positive solution of (3.1) belonging to C 2 (ft)nC(ft), and assume that f is Lipschitz continuous. Then u is symmetric with respect to x\, and uXl{x) < 0 for i e O with x\ > 0. Proof. For convenience of notation, we write x = {x\,y). We will show that uXl(x) > 0 if x = (xi,y) € ft satisfies xx < 0, and for (x\,y), {x[,y) G ft, u(xi,y)

if x\ < x[ < —x\.

(3.2)

If (3.2) is proved, then we fix x\ < 0 and let x[ —> —x\, and we deduce u{xi, y) < u(-xi,

y) if xi < 0.

Since we may prove the same for u(—x\,y), the above inequality implies u(xi,y) — u(—xi,y), i.e., u is symmetric in x\. As will become clear later, the proof of uXl(x) < 0 for xi > 0 also follows from (3.2). We now use the moving plane method to prove (3.2). Let a > 0 be given by -a — iaixenxiF ° r - a < A < 0, let T\ denote the plane x\ = A

The Moving Plane Method

19

and define S(A) = { i e f i : i i < A } . For x = {x\,y) £ £(A), set ux(x\,y)

= u(2X — x\,y),

w(x,A) = ux{x) — u(x).

We easily see that -Aux

= f(ux)

in E(A).

Since / is Lipschitz continuous, we can write f(ux(x))

- f(u(x)) = c(x,X)[ux(x)

-

u(x)\,

where c(x, A) is some bounded function for x € E(A), A e i J 1 . We can find some constant 6 > 0 such that \c(x, A)| —a is close to —a, then S(A) is narrow in the xi-direction, and by Theorem 2.6, the maximum principle can be applied to —Aw = c(x, \)w in S(A). On <9£(A), clearly w(x) is nonnegative and not identically 0. Hence the maximum principle infers that w > 0 in S(A). Let (j, := sup{A < 0 : w > 0 on S(A') for all - a < A' < A}. We want to show that fi = 0. We suppose \i < 0 and argue by contradiction. By continuity, w(x,fj,) > 0 in £(//). Since w is not identically 0 on 3S(,u), by the usual maximum principle or Harnack inequality (see Theorem A.42), w > 0 in S(/u). We will show that for all positive small e, w(x, fJ. + e) > 0 in T,(fi + e). Due to the definition of /z, this would give us the desired contradiction. By Theorem 2.6, we can find 6 > 0 small so that the maximum principle holds for Lu := An + c(x, X)u over any D C D, provided that \D\ < S. We now choose a closed set K in S(/x) such that |S(/i) \K\ < 5/2. Clearly, by compactness, w(x, ju) > 0 for x e K. Hence, by continuity we can find eo > 0 small so that w(x, fi + e) > 0, \E((j, + e)\K]<5\/xeK,\/e€

(0, e 0 ].

In E := S(/i + e) \ if, u; = w(x, /z + e) satisfies -Aw = cw, and on <9£, w is nonnegative and not identically 0 (since u = 0 o n d$l, and w = 0 on T M+£ , and w > 0 on d i Q . Therefore we can apply Theorem 2.6 to conclude that

20

Maximum

Principles

and

Applications

w > 0 in E. Therefore w > 0 in E(/x + e), as we wanted. This proves that /j, = 0, and (3.2) thus follows. Finally we show that uXl(x) > 0 if x = (xi,y) G Q is such that xi < 0. Since for any fixed A e (—a,0), w(x, A) > 0 in E(A) and w;(a;,A) = 0 on T\, we can apply the Hopf boundary lemma (see Lemma A.35) to conclude that wXl(x, A) < 0 when x € fi and xi = A. Since u>Xl(:r, A) = — 2uXl(x) when £i = A, we obtain uXl > 0. • If fl is a ball, then by Theorem 3.2 we know that a positive solution of (3.1) is symmetric about any plane passing through the center of fi. Therefore, we have the following result. Corollary 3.3 If H is a ball and f is Lipschitz continuous, then any positive solution of (3.1) is radially symmetric. Remark 3.4 (i) Corollary 3.4 is not true for solutions of (3.1) which are not positive. For example, when Q is a ball, many eigenfunctions of —AM

= \u in fi, u = 0 on dVi

are not radially symmetric (though the one corresponding to the smallest eigenvalue does not change sign and hence must be radially symmetric). (ii) This corollary is also untrue if the Dirichlet boundary condition is replaced by other boundary conditions. A well-kown result of Ni and Takagi [Ni-Takagi(1993)] shows that the problem —Au = A(MP — u) in fi, Dvu = 0 on <9f2, has a positive solution u which has a unique maximum point xo € dn, provided that 1 3 denotes the dimension of the bounded smooth domain fi. (iii) By using the moving plane method and also a "rotating plane" method, it was proved in [Lin-Tagagi(2001)] that, if fi is a ball, then the positive solution in (ii) above (called the "least energy solution") is symmetric in any plane passing through the center of fl and XQ. Theorem 3.2 can be applied to domains with "corners". For example, if Q is a cube, then we can use Theorem 3.2 to conclude that any positive

The Moving Plane Method

21

solution of (3.1) is as symmetric as Q. So if 0 = {{x\,X2) Q R2 : \xi\ < 1, \x2\ < 1}, then any positive solution u of (3.1) is symmetric with respect to x\ = 0, to X2 = 0, to X\ + X2 — 0 and to x\ — X2 = 0, which in turn implies that u is invariant under rotations around the origin of degree 7r/2 and integer multiples of n/2. An interesting example is when ft. is the "star of David" in R2 (see Fig. 1 (a)). x2

O W m(a)

(b)

(c)

Fig.l Star of David

Let us use D to denote this particular f2. We easily see that D is Steinersymmetric with respect to any straight line passing through the center and an outward-pointing corner point. But it is not Steiner-symmetric with respect to any straight line passing through the center and an inwardpointing corner point. As a result, we can use Theorem 3.2 to see that any positive solution of (3.1) with fi = D is invariant under rotations around the center of D of degree 27r/3 and integer multiples of 27r/3 (see Fig.l (b)). But this does not reflect the full symmetry of D, as clearly D is invariant under rotations around its center of degree 7r/3. This problem was considered recently by Kawohl and Sweers [Kawohl-Sweers (2002)], who used a "sliding method" (another well-known technique of using the maximum principle) to show the following result. Proposition 3.5 Any positive solution u G C 2 (Q) n C(f2) of (3.1) with Q, — D is invariant under rotations around the center of D of degree 7r/3.

Maximum

22

Principles

and

Applications

Proof. Let us choose the coordinates as in Fig. 1 (c). Moreover, we use /i, I2 and I3 to denote the straight lines X2 = 0, \f3x\ — X2 = 0 and \/3x\ + X2 — 0. As indicated in Fig. 1 (c), we denote by D\ the part of D that is between l\ and l2 whose point has positive coordinates. Suppose that u G C 2 (fi) nC(fi) is a positive solution of (3.1) on D, and v is obtained by a rotation of M around the origin of degree 7r/3: v{x) = u(Rw/3x),

Re(xi,X2) = {xi cos6 + 2:2 sin #,2:1 sin# + 2:2 cos^).

Since D is Steiner-symmetric in the directions of l\, I2 and Z3, by Theorem 3.2 we know that u is symmetric with respect the the lines perpendicular to, respectively, h, h and ^3. Moreover, uXl(x\,X2) < 0 for (2:1,2:2) G D\. Similarly, vxl(xi,X2) < 0 for (xi,x2) G D\. Let us also note that the above symmetries of u imply that u{x) = v(x) for x G l\ UI2 U /3. We want to show that u = u on D\. This would imply u = v in .D. We use a sliding method. Let us denote by a the length of the side of D\ which lies on l\. For t G [0,a], set D\ := {(2:1,0:2) : (2:1 +t,X2) G I?i} and v*(2:1,12) = v(xi +t,x2),

(xi,x2)

G D\.

Clearly -Aw* = / V ) in D\. We set to show that v* £ for all £ G [0,a]. For t < a but close to a, D\(t) := Di D £)j is a narrow set, and if/(2:) := u(x) — ^'(2:) satisfies - A w ' = c(2:,t)w/ in D(t), w* > 0 on dD(t), where c(x, t) is a bounded function and we have used the facts that u = v on Z2 and both w and v are decreasing in x\ in D\. Since w* is not identically 0 on dD(t), we can use Theorem 2.6 to conclude that w* > 0 in .D(i). Let to = inf{* G (0,a] : io*(a;) > 0 in D(s)Vs G [t,a]}. The above argument shows that to < a. If to > 0, then by continuity we obtain wto(x) >0 in D(to)- Moreover, - A w ' ° = c(x,t0)wto

in D{t0),

and wto is nonnegative and not identically 0 on dD(to). Therefore by the usual maximum principle or Harnack inequality, wto > 0 in D(t0). We now show that we can further "slide" v* under u to the right slightly, that is, there exists eo > 0 small such that wto~€ > 0 in D(to — e) for all e G (0,eo]. This would contradict the definition of to a n d hence prove

The Moving Plane Method

23

our claim that to = 0. Again we will use Theorem 2.6. Let 6 > 0 be small enough so that this theorem applies to Lu := Au + c(x, t)u when the underlying domain has measure less than 5. Then choose a closed set K in D(t0) such that |£>(t0) \K\ < 5/2. By compactness, wto~e(x) > 0 for x G K and all small e > 0, and \D(to — e) \ K\ < 5 for all such e. In D := D(to -e)\K, w(x) = wto~€(x) satisfies - A u ; = cw, and on dD, w is nonnegative and not identically 0. Therefore w > 0 in D by Theorem 2.6. Hence w > 0 in D(to — e), as we wanted. This proves to = 0 and thus v < u inDi. We can interchange the position of u and v in the above argument and hence deduce u < v in D\. Therefore u = v in D\. This finishes the proof. • Note that in the above proof, for the sliding method to work, it is important that we can first use Theorem 3.2 to guarantee that u = v on dD\ and both u(x) and v{x) are decreasing in x\ for x € Di. The symmetric domains in Fig. 2 below do not have any Steiner-symmetries and hence the above method does not work for them. Do positive solutions of (3.1) over these domains possess the symmetries of the domain? Kowohl and Sweers in [Kawohl-Sweers(2002)] conjecture that they do, but no proof (or counter example) is known yet.

Fig. 2 Symmetric domains without Steiner-symmetry

3.2

Symmetry over the entire space

The moving plane method was used in [Gidas-Ni-Nirenberg(1981)j to study the radial symmetry of positive solutions of the following entire space prob-

24

Maximum

Principles

and

Applications

lem: -AM

= f(u) in RN {N > 2),

lim u(x) = 0.

(3.3)

|x|—>oo

The following result improves some results in [Gidas-Ni-Nirenberg(1981)] and is proved by Li and Ni [Li-Ni(1993)]. T h e o r e m 3.6 Suppose that f(s) is Lipschitz continuous and nonincreasing for sufficiently small s > 0. Then any positive solution u € C2(RN) of (3.3) is radially symmetric about some XQ G RN and ur < 0 for r = \x — XQ\ > 0. Proof. Let x = (xx,y) be an arbitrary point in RN. Its reflection with respect to the hyperplane T\ := {x G RN : Xi = A} will be denoted by x*, namely, xx = (2A — X\,y). Denote Y,\ := {x € RN : x\ < A}. We observe that, for A > max{xi,0}, -A,

,„, _ 4A(A-a; 1 )

l»1-N= \x\ ,J.+ \x* J Let u € C2(RN)

>0-

(3-4)

be a positive solution of (3.3) and define

A := {A G R1 : u(x) > u{xx) Vx G E A , uXl(a:) < 0 Mx G T A }. By our assumption on f(s) we can find ro > 0 such that f(s) is nonincreasing for s G (0,ro]. Since u(:r) —> 0 as |rr| —> oo, there exist Ro < R\ such that Ro > 1/ro, max u(x) < ro, max u(x) < mo := min u(x). |x|>fi 0

\x\>Ri

(3.5)

\x\
We divide the following discussion into four steps. S t e p 1. We show that [i?i, oo) C A. Indeed, in view of (3.4), for A > R\, w{x) := u(x) — u(xx) satisfies, for xeZx\BRo, where BR = {x£RN : \x\ < R], —Aw = c(x)w in

SA

\

BR0,

w = 0 o n T\,

lim w(x) = 0, |a;|—*oo

where c{x) is a bounded non-positive function. Moreover, since A > R\ and x\ < A, we have |x A | > R\ and hence by (3.5) we find w(x) > 0 on ~BRa n S A .

(3.6)

The Moving Plane Method

25

Therefore we have —Aw = c(x)w in

SA

\

BR0,

W

> 0 on d(Y,\ \ BRo),

lim w(x) = 0. |x|—»oo

By the strong maximum principle (see Theorem A.36) we deduce w > 0 in SA \ BR0 . Then the Hopf boundary lemma and the Harnack inequality imply w > 0 in EA \ BRo and wxi < 0 on Tx. In view of (3.6), we have proved w(x) > 0 in SA for all A > R\. Hence [i?i,oo) C A. S t e p 2. We prove that A is an open set in (0, oo). Let A0 £ A n (0, oo). We want to show that (A0 - e, A0 + e) C A n (0, oo) for all small e > 0. Since we already knew that [Ri,oo) C A, we may assume that Ao G (0, i?i]. It follows from the assumption Ao € A that u(x) - u(xx°) > 0 in SA 0 , uXl < 0 on TXo.

(3.7)

By continuity, we can find e\ > 0 small such that uXl (x) < 0 if |a;| < i?i + 1 and A0 - 4ei < x: < A0 + 4ei.

(3.8)

It follows that, for any A £ (Ao — ei, Ao + ei), u(x) > u{xx) uXl < 0

in {x G S f i l + i : A0 - 2ei < xi < A}, on TxnBRl+1.

. >

Denote 6 = min{u(a;) - u(xx°) : -(Rr

+ 1) <

Xl

< A0 - 2ei, |x| < Ri + 1}.

By (3.7), S > 0, and hence u(x) - u{xx) > 0 in {x G 5 R l + 1 : -(Ri

+ 1) < xx < A0 - 2ei}

(3.10)

for any A G (Ao — e, A + e) if we take e G (0, e\) small enough. Combining (3.9) and (3.10), we obtain, for A € (A0 - e, A0 + e), u{x) > u(xx) in B~Rl+i n E A , uXl < 0 on TxnBRl+i.

(3.11)

Now for A G (Ao — e, Ao + e), define w{x) = u(x) — u(xx) and we find that w jk 0 in SA \ -B_Rj+i and similar to before —Aw = cw in SA \ BRl+i,

w > 0 on C>(SA \ Bfl 1 + i),

lim w(a;) = 0. \x\—*oo

Maximum

26

Principles

and

Applications

By the choice of i?i, we know c(x) < 0 in SA \-Bfli+i- Hence by the strong maximum principle, for A G (Ao — e, Ao + e), w > 0 in

EA

\BRl+1,

wXl < 0 on T\

\-BR1+I,

that is, u(:r) - u(a;A) > 0 in EA \ BRl+1,

uXl < 0 on T\ \

BRl+i.

Together with (3.11), this proves (Ao — e, Ao + e) C A, as we wanted. Step 3. Either A n (0, oo) = (0,oo) or u{x) = u(xXl) for some Aj > 0. Let (Ai, oo) be the component of the open set A n (0, oo) containing (i?i,oo). By the continuity of u we have w{x) = u(x) - u(xXl) > 0 Vx 6 E A l . Moreover, —Aw = cw in

SAJ

, w = 0 on T\1,

lim w{x) = 0. \x\—+oo

Since w > 0 in S A l , though we do not know whether c is non-positive, we can apply the Harnack inequality to conclude that either w = 0 in T,\1 or w > 0 in S A l , and in the latter case, by the Hopf boundary lemma, wXl < 0 on T\ x . But then we find Ai G A in the later case. By Step 2, this is possible only if Ai = 0 . This finishes our proof of Step 3. Step 4. Completion of the proof. If u{x) = u(xXl) then u is symmetric with respect to the plane T\lt and since (Ai,oo) c A, we know that uXl < 0 for x\ > \\. If the other alternative occurs in Step 3, then other u(x) = u(x°) or u(x) > u(x°). In the former case u is symmetric with respect to To and as before uXl < 0 for xi > 0; in the latter case, we must have uXl < 0 on To and we can apply the argument in Step 3 to w(—x\, y) to deduce that for some A2 < 0, u(x) = u(xX2) and uXl > 0 for xi < A2. Hence u is always symmetric with respect to some T\, and is strictly decreasing away from T\. Since (3.3) is invariant under rotations, we may take any direction as the xi-direction and conclude that for any given direction, u is symmetric with respect to some hyperplane T perpendicular to that direction, and is strictly decreasing away from T. This implies that u is radially symmetric about some point XQ in RN and ur < 0 for r — \x — x0\ > 0. The proof is complete. • An important example covered by Theorem 3.6 is f(s) = sp — s, p > 1. However, Theorem 3.6 excludes important functions like f(s) = sp for s > 0

The Moving Plane Method

27

and p > 1. The following result, whose proof can be found in [Gidas-NiNirenberg(1981)] (see also [Fraenkel(2000)]), covers these cases. Theorem 3.7 Suppose that u £ C2(RN) is a positive solution of (3.3), N > 3 and u(x) = 0(|a;|~ m ) at infinity for some m > 0. Suppose further that (i) for s £ [0,uo], where UQ — maxflw u(x), f(s) = /i(s) + /2(s) with fx Lipschitz continuous and fi continuous and non-decreasing, (ii) for some a > max{(iV + l)/ro, (2/m) + 1}, f(s) = 0(sa) near s = 0. Then u(x) is radially symmetric about some point XQ G -R^, and ur < 0 for r = |x — XQ\ > 0. Moreover, there exists some k > 0 such that

lim l a r l " - 2 ^ ) = A;. |x|—»oo

A well-known example covered by Theorem 3.7 is / ( s ) = s(^+2)/(iV-2) In this case, we can take m = N - 2 and a = (N + 2)/(N — 2). Therefore, any positive solution of - A « = u(^+a)/(Ar-2) in RN 22 - ^ „ „ l „ l

u{x) = 0(\x\ ~

(N

>

3)

. _

) as \x\ —* co

(3-

1 2

)

must be radially symmetric about some XQ € RN • Let us note that all such solutions are explicitly known (see [Cerverno-Jacobs-Nohl(1977)]), and they are given by i(3!)=

(/+lx-xo|0

'A>M0€"

In [Chen-Li(1991)], by making use of the Kelvin transformation, the condition u(x) = 0(\x\2~N) as |x| —> oo in (3.12) was removed. We will discuss this in detail in Section 7.4 later. Further related results can be found in [Chen-Li-Ou(2005)], [Chen-Li-Ou(2003)], [Busca-Manasevich(2002)] and the references therein. R e m a r k 3.8 The moving plane method can also be applied to certain so called cooperative systems of elliptic equations and to deduce symmetry of the positive solutions; see [Troy(1981)], [de Figueiredo(1994)], [BuscaSirakov(2000)] and [Busca-Manasevich(2002)].

Maximum

28

3.3

Principles

and

Applications

Positivity of nonnegative solutions

The symmetry results in Sections 3.1 and 3.2 are valid for positive solutions. In many applications, the natural solution is nonnegative. Then one is interested to know whether such a solution is strictly positive; if it can vanish in a set of positive measure yet not identically zero, such a vanishing set is often called the dead core of the solution. Consider now the problem - A M = f(u) in

ft,

(3.13)

where ft is a domain in RN (not necessarily bounded). If f{u) is Lipschitz continuous and /(0) > 0, then f(u) > f{u) — /(0) > — Cu for some positive constant C, and we obtain, for any nonnegative solution u of the above problem, Au - Cu < 0 in ft. By the strong maximum principle (see Theorem A.38), we deduce either u = 0 in ft or u > 0 in ft. The strong maximum principle has been generalized to various nonlinear equations. The following well-known result is due to J.L. Vazquez [Vazquez(1984)] (with the converse part from [Benilan-BrezisCrandall(1975)]); see [Pucci-Serrin(2004)] and the references therein for further discussions. Theorem 3.9

Ljocity

Let u £ Lj0C(Q) be such that u > 0 a.e. in ft, Au €

and

Au < (3{u) a.e. in {x e ft : 0 < u(x) < a}, where a is a positive constant and /3 : [0, a] —> R1 is a continuous nondecreasing function with /?(0) = 0. / / either /3(d) = 0 for some 6 > 0 (and hence (5{t) = 0 on [0,6] due to the monotonicity), or (5{t) > 0 in (0, S) C (0, a) and 1-1/2 Jo

L

Jo

0{t)dt

ds = oo,

then u is either identically 0 in ft or strictly positive in ft.

(3.14)

The Moving Plane Method

29

Conversely, i//3(0) = 0, (3(t) > 0 in (0,6) for some 6 > 0 and (3.14) not satisfied, i.e.,

is

T-l/2

! \ f'mdt

as < oo, Jo l Jo then for every XQ G R N and every R > 0, there exists a function u € Cl{RN) with the properties that u > 0 , ^ 0, Au G L°°(RN), Aw = /3(u) ami w = 0 on RN

\

BR(XO).

From Theorem 3.9, we see immediately that if /(0) > 0, or /(0) = 0 and f(u) > —/3(u) for w > 0 small, where /? is a continuous non-decreasing function satisfying the conditions in the first part of Theorem 3.9, then any nonnegative solution of (3.13) is either identically 0 or strictly positive in When /(0) < 0, it turns out that, even if f(u) is very smooth, (3.13) may have nonnegative solutions which is not identically 0 but vanishes at certain interior points in fi. For example, the one dimensional problem —u" = u — 1 has a nonnegative solution u = 1 — cos a; that vanishes at x = 2kir, k = 0, ± 1 , ±2,.... More generally, consider -u" = f(u) in (0, a), «(0) - u(a) = 0,

(3.15)

where / : [0, oo) —* R1 is a continuous function such that for some constant

P>0, (i) f(s)<0\/se[0,(3), /(/?) = 0; (ii) / ( a ) > 0 V s G ( / ? , o o ) ; (iii) f™ f(s)ds = +oo. Denote F(s) = f* f(t)dt. From the above assumptions on / we easily see that there exists a unique 6 > (3 such that F(0) = F(9) = 0, F(s) < 0 Vs G (0,0). Moreover, [-2F(U)]-1/2^M
L:=

./o since -F(u) f°[-f(s)]ds

= fQu[-f(s)]ds > l-f(9)/2}(8

> [-/(0)/2]u for all small u > 0 and -F(u) - u) for all u < 9 with (9 - u) small.

=

Maximum

30

Principles

and

Applications

Let x = x(u) = / [-2F(s)}~1/2ds, Jo

0
Clearly x(u) is a C 1 function with positive derivative [—2F(u)]~1/2. Hence x(u) has an inverse function u : (0, L) —> (0,9) which is C2 in (0, L) and satisfies u'(x) = [~2F{u{x))]l>2, u"(x) = -f(u(x)),

x G (0,L).

It is easy to see that u can be extended to a C2 function over [0, L] with u(0) = 0,u(L) = 0, M'(0) = u'(L) = 0. Since u'(L) = 0 we can extend u to (L,2L] by reflection about x = L: u(x) = u(2L — x) for x G {L,2L\. The extended M(X) satisfies -u" = f(u) in (0,2L), u{0) = u(2L) = 0, u'(0) = u'(2L) = 0. We can further extend u to (2L,4L) by reflection about x = 2L, and the extended u(x) satisfies -u" = f(u) in (0.4L), «(0) = u ( 4 l ) = 0, and u(2L) — 0, u > 0 in (0, 2L) U (2L, 4L). This gives an example where a nonnegative solution of (3.15) (with a — 4L) vanishes at certain point inside the domain but is not identically 0. Surprisingly, it has been shown by A. Castro and R. Shivaji [CastroShivaji(1988)] that examples like the one above can only be found in dimension 1! More precisely, if the interval (0, a) is replaced by a high dimension ball B, the corresponding problem of (3.15) in B cannot have a nontrivial nonnegative solution that vanishes somewhere in B. Theorem 3.10 If £1 is a ball of dimension no less than 2, f(u) is locally Lipschitz continuous on [0, oo) and /(0) < 0, then any nonnegative solution of (3.13) must be positive. (Note that 0 is no longer a solution due to

/(o) < o.; Proof. The theorem will be proved by making use of the moving plane method. For each given direction v in RN, i.e., a vector u £ RN with \v\ = 1, and each A G R1, we define a{u) = inf x • v\ Tx = {x G RN : x • v = A}; fi^ = {x G fi : x • u < A}.

The Moving Plane Method

31

Therefore Tx is a hyperplane in RN perpendicular to v, Qx is a part of fi lying on one side of Tx, and when A = a(i/), then Tx touches Ct at exactly one point on its boundary. Let R1^ be the reflection map in RN in the hyperplane Tx: R\{x) = x + 2(A - x • v)v, x e RN. We will use the notation

x\ = Rl(x), (niY = RWJ. Under the above assumptions, if A — a(v) is positive and small, (fi^)' fl. We now can define

c

\*(u) = sup{/i > a{v) : (Clx)' C Q for every A < /i}. We easily see that A*(v) can be characterized as the unique value of A such that Tx passes through the center of Q. Now as in the proof of Theorem 3.2, by comparing u(x) with u\(x) := u(xvx) over Qux for a(v) < A < \*(v), and making use of Theorem 2.6 and the usual weak maximum principle, we deduce that uvx(x) > u(x) on Q^. It follows that u(x) is non-decreasing in the direction v in the half ball fiw„NSuppose by contradiction that u(x§) = 0 for some XQ £ Cl. Then the above proved mono tonicity property of u(x) implies that u(x) = 0 on the line segment connecting XQ to a boundary point y 6 dCl such that the vector yxo is in the same direction as some v satisfying xo G fiw„\- All these line segments form a cone K mil with vertex XQ. (In fact, if we choose the coordinates such that ft = -B^(O) and XQ = (a, 0, ...,0) with 0 < a < R, then K = {x = (x\, ...,XN) £ BR(0) : X\ > a}.) Therefore u is identically 0 in K, and hence Aw = 0 in K, which is a contradiction to the assumption that /(0) < 0. • The above proof is modelled on the one given in [Damascelli-PacellaRamaswamy(2003)], and can be extended to bounded domains more general than a ball; see [Castro-Shivaji(1988)] and [Damascelli-PacellaRamaswamy(2003)] for more general results. However, it is unknown so far whether Theorem 3.9 holds for any bounded smooth domain in RN with N > 2. R e m a r k 3.11 The moving plan method has been extended to some quasilinear elliptic problems; see [Damascelli-Pacella(1998)], [DamascelliPacella-Ramaswamy(2003)], [Damascelli-Sciunzi(2004)] and the references therein.

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Chapter 4

The Method of Upper and Lower Solutions The "upper and lower solution method", also known as the super and subsolution method, is a basic tool in nonlinear partial differential equations. In this chapter, we discuss some classical and weak versions of this method. The weak version is often more convenient to use, though it has some structural requirements for the elliptic operator which do not appear in the classical version.

4.1

Classical upper and lower solutions

Let L, ft and B be as in section 2.1, namely, Lu = alj(x)DijU + bl(x)DiU + c(x)u has Ca(£l) coefficients and is strictly uniformly elliptic in the bounded domain CI which has C2'a boundary <9fi; To and Ti are two disjoint open and closed subsets of d£l with To U Ti = dQ, and _ fu on To, \ Duu + au on T\, where a 6 C 1,Q (Q) is nonnegative, and v is the outer normal to Y\. It is possible that either To or Ti is the empty set. By Theorems 2.1 and 2.4, the principal eigenvalue Ai = Xi(L,B,Q,) exists and the strong maximum principle holds for (L, B, Q) if and only if Ai > 0 . We now consider the elliptic boundary value problem —Lu = f(x, u) in Q, Bu = 4>(x) on dfl, 33

(4-1)

Maximum

34

Principles and

Applications

where / G Ca(fl x / ) , i" is some finite interval in R1, (f>\r0 G C 2 ' a (ro) and Definition 4.1 A function u 6 C 2 ' a (f2) is called an upper solution to (4.1) if it satisfies —Lu > f(x, u) in ft, ^ u > ^ on <9fi. It is called a lower solution of (4.1) if the inequalities above are reversed. Theorem 4.2 Suppose that v, u> G C2'a(fl) are lower and upper solutions to (4-1), respectively, and v < w in fl. Moreover, suppose that f G Ca(fl x / ) satisfies, for some m > 0, / ( x , s) - / ( x , t) > - m ( s - J ) V i e n , s , t 6 [v(x), w(x)], s > t,

(4.2)

where I = [ram.^v{x),m.&x^iui{x)\. Then (4-1) has at least one solution u satisfying v max{m, maxjy |c(x)|}, then, with UQ = v, un converges from, below to a solution u* of (4-1) and with UQ = w, un converges from above to a solution u* of (4-1)- Furthermore, the convergence of {un} holds in C2(fl), and any solution u of (4-1) with v 0, the strong maximum principle holds for (L*,B, fl). With u0 — v, we find, —L*(u\ — M0) > 0 in fl, B{u\ — u0) > 0 on dfl, and -L*(ui

- w) < 0 in fl, B(ui - w) < 0 on dfl.

Therefore it follows from the maximum principle that w > u\ > UQ in fl. Now for i > 1, due to (4.2) we have, inductively, —L*(ui+i —Ui)>0

in fl, B(ui+i

- ut) > 0 on dfl,

—L*(uj+i - w) < 0 in fl, B(ui+i

- w) < 0 on dfl.

and

The Method of Upper and Lower

Solutions

35

Hence w > ui+i > m in 0 . Therefore {un} is an increasing sequence of functions bounded from above by w. We may define «» := limn^oo unSince {m*un + f(-, un)} is a bounded sequence in L°°(fl), by the Lp theory (see Theorems A.28 and A.29), {un} is a bounded sequence in W2,P(Q.) for any p > 1. By the Sobolev imbedding theorem (Theorem A.62 and Remark A.18), it follows that {un} is bounded in C 1 (fi), which in turn implies that {m*un + f(-,un)} is bounded in C a (ft). Therefore by the Schauder theory (Theorem 4.4), we find that {un} is bounded in C2'a(Q) and hence it has a convergent subsequence in C2(fi). As we already know that un converges to u , point-wisely in Q, we must have u„ —• u» in C 2 (fi). So we can now pass to the limit in (4.3) to obtain —Lu* + m*u* = f{x, u*) + TO*W* in fi, Bu» = <^> on <9fi. That is, w* is a solution to (4.1). Moreover, from w > un > v, we find w > u» > v. If M is any solution of (4.1) satisfying w > u > v, then we can replace w in the above discussion by u and obtain u > un > v and hence u > u*. This shows that u* is the minimal solution of (4.1) in the order interval [v,w\. If we take UQ = w, we can analogously show that un decreases to a maximal solution u* in the order interval [v,w], and the convergence holds in C 2 (fi). We leave the details to the interested reader. D Condition (4.2) is always satisfied if f(x, s) is Lipschitz continuous in s. However, this condition is not necessary for the existence of a maximal and minimal solution in [v, w]. The following result is essentially due to H. Amann [Amann(1971)], where condition (4.2) is not needed. Theorem 4.3 Suppose that v, w G C2'a(Q) are lower and upper solutions to (4-1), respectively, and v <w in Q. Moreover, suppose that f £ Ca(fl X I) with I = [minQv(x),maXftW(x)]. Then (4-1) has a minimal solution M» and a maximal solution u* in the order interval \v,w\. If furthermore, c(x) < 0 and f(x, s) is non-increasing in s for s e / , then u , = u* and hence (4-1) has a unique solution in [v,w]. Proof. Firstly we may assume c > 0 in Q, for otherwise, we can replace L by L + rn and f(x, s) by f(x, s) + ms with some large positive m, and all the conditions in the theorem are retained (except for the uniqueness part, which will be treated separately).

36

Maximum

Principles

and

Applications

Since f(x, s) is Holder continuous, there exists some 7 > 0 such that \f(x, s) - f(x, t)\ < 7 k - t\a Vx e H, s, t G / .

(4.4)

We now divide our discussions below into several steps. Step 1. If f(x, s) is non-increasing in s for s £ I, and c(x) < 0 in fi, then (4.1) has at most one solution in [v,w] Let u\ and u2 be solutions of (4.1) in [v, w]. Set Qi := {x 6 fl : «i(a;) > ^ ( z ) } - We show that fii must be empty. Otherwise, it is an open set, and by the monotonicity assumption on / , we have —L(u\—u2) < 0 in ili, &(ui—U2) = 0 o n diliDdil,

ui—n?, — 0 on SfiiDfi.

Therefore, by the maximum principle (applied to every component of Qi), we obtain ui < u2 in fii, contradicting its definition. Hence Qi is empty and ui < 112 in Q,. We can similarly prove that 112 < ui- Therefore wi = ^2Step 2. If w, w S C 2 ' a (fi) are lower and upper solutions of (4.1) and v < w < w < w, then the problem -Lu — f(x, w(x)) — 7(14 — w(x))a in fl, Bu = <j> on dVL

(4.5)

has a unique solution w* in the order interval \w, w], and the problem -Lu = f(x, w(x)) + -y(w(x) - u)a in fl, Bu = on dQ,

(4.6)

has a unique solution w* in [w,w]. By Step 1 we know that each of (4.5) and (4.6) has at most one solution in [tD,w]. We will use Theorem 4.2 and a perturbation argument to show that (4.5) has a solution in [w, w]\ the proof for (4.6) is similar. Let e(x) be the unique solution of —Lu = 1 in il, Bu = 0 = 0 on IV Then by the strong maximum principle, we know that e(x) > 0 on Q. Now for every positive integer n, we define wn = w — (l/n)e and u-w(x)-\ where e > 0 is so small that 1/n > 7(—e(x)) .

e(x)\

,

The Method of Upper and Lower

Solutions

37

Then we easily checks that —Lwn < fn(x,wn)

in fi, Bwn < on dfi,

and by (4.4), —Lw > fn{x, w) in fi, Bw > <> / on Oil,. Let us observe the important fact that f„(x, s) is Lipschitz continuous in s when s€

[wn(x),w(x)].

Therefore we can use Theorem 4.2 to conclude that the problem —Lu = fn(x, u) in fl, Bu = on dfl

(4.7)

has at least one solution wn in [«;„,«)]. By Step 1, we know this solution is unique. Since / n + i(:r,u; n +i(a;)) > fn(x,wn+1(x)), and wn+1 > u)„ +1 > wn, we can apply Theorem 4.2 for the pair of lower and upper solutions {wn, wn+i) to (4.7) to conclude that it has at least one solution w' in [wn, w n +i]. Clearly w' is also in the order interval {w„,w]. Therefore by the uniqueness of wn we necesarily have w' = wn. It follows that wn < wn+i < w for all n. Let w* = lim n _ooW n . Then w < w* < w. Moreover, by the Lp theory, the Sobolev imbedding theorem and the Holder theory, we find that {wn} is bounded in C 2 , a (fi) and hence has a convergent subsequence in C2(Q). As we already know that wn —> u>* point-wisely, we must have wn —• u>» in C 2 (fi). Now we let n —> oo in (4.7) and find that w* is a solution to (4.5). The proof for the existence of a solution w* to (4.6) is analogous, where we replace wn by wn = w + (\/n)e and replace fn(x, s) by gn(x, s) = f(x, w(x)) + 7 f w(x) A

e(x) - sj .

The detailed argument is left to the interested reader. S t e p 3. Completion of the proof. In Step 2, we take, inductively (w,w) = (vn,w), where v0 — v and vn is the unique solution of (4.5) with w = v n - i - This is possible since by (4.4), we find that each vn is a lower solution of (4.1). Clearly v < vn-\ < vn < w. Let M» = limn-tooVn. Then as before, we can apply the Lp theory, the Sobolev imbedding theorem and the Holder theory to conclude that vn converges in C2(Q) to u„, and u* is a solution of (4.1) in the order interval [v, w]. We claim that w« is the minimal solution. Indeed, if u is any solution

38

Maximum

Principles and

Applications

of (4.1) in [v,w], then in the above argument, we replace w by u and we find that the vn is not changed and v < vn < u. Therefore u„ < u. We leave the proof for the existence of a maximal solution u* to the reader. • Remark 4.4 In [Amann(1971)], the case that 4>\ri depends on u was also discussed, where stronger smoothness conditions on the coefficients of L, etc. are needed. Consider next the problem -Au = f(x, u, Du) in ft, Bu = 0 on dfl,

(4.8)

N

where ft is a bounded domain in R as in Theorem 4.2 except that we only require dQ, to be C 2 , B is also as in Theorem 4.2 except that we only require a G C ^ T i ) , Au = ali (x)DijU, with aij e C(ft), oiJ' = a^ and a tf (a:)6& > 0 Vz € ft, £ e i? W \ {0}. Suppose that p > N. Then by the Sobolev imbedding theorem, W2'p(fl) is imbedded in C ^ Q ) . We say w G W2-P(Q) is a solution to (4.8) if -Au = f(x, u, Du) a.e. in Cl and Su = 0 on <9fi. A function u € W 2 ' p (f2) is called an upper solution to (4.8) if —Au > f(x, u, Du) a.e. in ft and Bu > 0 on 9ft. It is called a lower solution if the above inequalities are reversed. The following result is due to Amann and Crandall (see [AmannCrandall(1978)] Theorem 1 and the remark on page 787). Theorem 4.5 Let f : Q x R1 x RN —> R1 be a continuous function such that it is locally Lipschitz continuous in £ and n, where (x, £, 77) denotes a generic point of D, x R1 x RN. Assume moreover that \f(x,t,v)\<

c(|e|)(l + \v\2) V(x,tv)eUxR1x

RN,

for some increasing function c : [0,00) —> [0,00). Let v be a lower solution and w be an upper solution of (4-8) such that v < w in Q,. Then (4-8) has a minimal solution w, and a maximal solution u* in the order interval [v, w]. Remark 4.6 The boundary condition Bu — 0 in (4.8) looks less general than that in (4.1). But the inhomogeneous boundary condition can usually be transformed into a homogeneous one under suitable changes of variables of the form v — u — tp where t/j satisfies BI}J = <\>.

The Method of Upper and Lower

4.2

39

Solutions

Weak upper and lower solutions

Let fl be a bounded domain in RN with boundary dfl, and let A be a second order quasilinear elliptic operator in divergence form: ,Du(x))^

for a.e. x £ fl,

(4.9)

where Ai (i = 1,..., N) satisfies the following conditions: (Al) Each Ai : fl x Rx x RN —» R1 satisfies the Caratheodory conditions, i.e., Ai(x, t, £) is measurable i n a ; g f i for all fixed (t, £) £ R1 x RN, and continuous in (£, £) for a.e. fixed a; G fi. (A2) There exist constants q £ (l,oo) and co > 0 and a function ko £ Lq (fl) with q' = q/(q—l) such that, for i = 1, ...,N, for a.e. x £ fl

a n d ( t , 0 eR1

xRN,

\Mx,t,0\
+

\^-1).

(A3) For a.e. x £ fl, for all t 6 -R1 and £, f € i ^ with £ ^ £', E ^ ( ^ ( x , *, 0 - A ( x , *, £'))(& - £ ) > 0. (A4) For some a > 0 and &x G £ 9 ' ( ^ ) , for a.e. x £ fl and all (£,£) G

E£ 1 A i (x,t,0^>«l^-fci(a:)If i4u = S 5 = 1 D i ( o y ( i ) D J - u ) , then clearly (Al)-(A4) are satisfied with AiU = Ejl1al-7(:r).DjU and q = 2 provided that aij' G L°°(fi) and E& = 1 a y (a:)&$j > a|£| 2 V£ G i ? N and a.e. x £ fl. As a consequence of (Al) and (A2), the semilinear form a(u,v) = 'SiL1 I in

Ai(x,u,Du)Divdx

is denned on W1^(fi) x W ^ f t ) . We now consider the boundary value problem —Au + p(x, u, Du) = / in fl, u = g on dfi,

(4-10)

40

Maximum

Principles

and

Applications

where A is given by (4.9), p is a Caratheodory function, g G W 1 , 9 (fi), f G W~1,q'(Q,) := (W01,<7(fi))*, the conjugate space of the Banach space W0'q(Q). Here by u = g on d£l, we mean (u — g) G W 0 ' 9 (fi). We say that u is a weak solution of (4.10) if u £ W /1 ' 9 (fi), u = g on dfl, p(x,u(x),Du(x)) belongs to Lq (ft) and a{u,v)+

[ p{x,u,Du)vdx=(f,v)

Vwe ^ ' ' ( f i ) ,

where (/, v) denotes the pairing between W'1^'^)

and W01,9(ft).

Definition 4.7 A function u is called a weak upper solution of (4.10) if u G W1,q{Ct), u > g on dfi, p(x,u(x),Du(x)) belongs to Lq (fi) and a(u,v) + / p(x,u,Du)vdx

> (f,v)

W G Wo' 9 (fi) with v > 0 a.e. in fi,

where by u > g on 317, we mean (g — u)+ := max{g — u, 0} G W0 ' 9 (fi). It is called a weak lower solution if the inequalities are reversed. The theory for weak upper and lower solutions to be discussed below will be based on variational inequalities. The main abstract tool for the existence result will be the theory of monotone operators. We now recall a useful result for such operators. Let V be a reflexive Banach space whose dual space is denoted by V*. For u G V and / 6 7*, we denote by (f,u) the pairing between u and / . An operator A : V —> V* is called pseudo-monotone if un —> u weakly in V and liirtn^oo(^4wn, "n - u) < 0 imply (Au,u - v) < lim n _^ 00 ( J 4u„,u n - v) for all v G V. A : V —> V* is called bounded if it maps bounded sets in V to bounded sets in V*. We have the following well-known result (see Theorem 2.3 in [Showalter(1997)]). Theorem 4.8 Let V be a separable reflexive Banach space, K a closed, convex non-empty subset of V, A : V —> V* a bounded pseudo-monotone operator, and f G V*. Assume that there is a VQ G K and p > 0 such that (Av, v-v0)

> (/, v - v0) \/v G K, \\v\\ > p.

(4.11)

Then there exists a solution of the following variational inequality: ueK

: (Au, v - u) > (/, v - u), Vv G K.

(4.12)

The Method of Upper and Lower

Solutions

41

Let us note that if we take K = V in (4.12), then u is a solution to (4.12) if and only if Au = f. Moreover, with K = V, (4.11) is satisfied if A is coercive, namely, (Av, v) MHoo \\v\\ hm

——-n— = ° ° -

Theorem 4.9 Suppose that v and w are weak lower and upper solutions of (4-10), respectively, and v < w a.e. in ft. Suppose further that there exist a constant c\ > 0 and a function k\ € Lq (ft) such that \p{x,t,0\
+ Cl\^-1

(4.13)

for a.e. i e f i , all £ £ RN and all t € \v{x), w(x)}. Then (4-10) has a weak solution u satisfying v < u < w a.e. in fi. Proof. We first associate to problem (4.10) and the given functions v, w a coercive boundary value problem obtained by modifying the coefficient functions outside the interesting range {u : v < u < w}. We then show that any solution of the modified problem satisfies v < u < w and hence is a weak solution of the original problem. We divide the discussions to follow into several steps. S t e p 1. The modified problem. For i = 1,..., N, x G Q and (t,$) £ R1 x RN, let

{

Ai(x,v(x),£) Ai(x,t,£)

iit
Ai(x,w(x),£) ift>w(x). It is easily seen that Ai still satisfies (A1)-(A4). We will also need the truncation mapping T : W1,q(£l) —> W1'q(Q):

(

v(x) if u(x) < v(x), u(x) if v{x) < u(x) < w(x), w(x) if u(x) > w(x).

We want to show that the problem -Au + p{x, Tu, D{Tu)) = f in ft, u = g on dfl has a solution, where {Au)(x) =

Z?=1DXiAi(x,u{x),Du(x)).

S t e p 2. Further changes and modifications.

(4.14)

42

Maximum

Principles

and

Applications

In order to use abstract theory to prove the existence of a solution to (4.14), we now consider further changes. Let A!i{x,t,S)=Ai{x,t

+ g(x),S + Dg{x)),

p,(x,t,£)=p(x,t

+ g(x),S + Dg(x)).

We easily see that u is a weak solution to (4.14) if and only if u — g is a weak solution of -A'u + p'(x, Tu, D(Tu)) = / in fi, u = 0 on dQ,

(4.15)

where (A'u)(x) =

HtiDXlA'i(^u(x),Du{x)).

Let 7 : Q, x R1 —> R1 be given by [ - ( - * - g{x) + v(x)Y~l l{x, t) = I 0 { (t - w{x) + g{x))q-1

if t < v(x) - g{x), if v(x) - 3(2) w(x)-g(x).

We now define the semilinear form b by 6(u,^) = EJl 1 I A'i{x,u,Du)(t>Xidx+ Jn

[

p'(x,Tu,D(Tu))(j>dx

JQ

+P / ry{x,u)(j>dx, Jo. with u, 4> S W0 '*(fi) and some fixed constant /? > 0 to be determined below. For fixed u e Wo' 9 (fi), it is easy to check that A^(-,u,Du),

P'(;Tu,D(Tu)),

7(.,«)

e L*'(Q),

where q' = q/(q - 1). Therefore, £>(u, 0) is a bounded linear functional of <> / G W01,9(fi). Hence there exists a unique Fu G W~ 1 , 9 '(n) such that 6(u,0) = {Fu,4>)^ e ^ f i ) . S t e p 3. The operator F : W 0 1,9 ( fi ) ~» W~ 1,9 '(ft) is bounded, pseudomonotone and coercive. If u has a bound on its W 1,9 (fi) norm, say ||w|| < C, then it is easy to see that \b(u,|| for some Cx > 0 and all 0 G ^ ' ' ( f l ) . This implies that F is a bounded operator.

The Method of Upper and Lower

43

Solutions

To show that F is pseudo-monotone, we suppose that un converges weakly to u in WQ'9(Q) and lim„_ >00 (FM„,u n — u) < 0, i.e., lim n _ 0 0 6(u„,u n - u) < 0. Denote bi{u,4>) = Y$Ll / A-(:r,u,.Du)Xid:r, Jn b2{u,(fr)= / p'(x,Tu,T(Du))(/>dx

Jn

+ /3 / *y(x,u)(f>dx.

Jn

Since {un} is bounded in W0 ,q(0.), so is {Tu„}. Moreover, un —> w in L 9 (fi). Therefore, {p'(-,Tu n ,£>(Tu n ))} and {7(-,u n )} are bounded in L 9 '(fi). It follows that b2(un, un — u) —> 0 as n —» oo, which implies that lim„_oofri(wn, ti n - u) < 0.

(4-16)

Since A' satisfies (Al) — (A4), by a well-known result the operator F\ : W01,9(fi) -» W - 1 - ? ' ^ ) defined by b1(u,tf>) = (F1u,) \/cf>eW^(Q) is pseudo-monotone (see Theorem 6.1 in Chapter II of [Showalter(1997)]). Therefore (4.16) implies that h(u,u-v)<\jmn^00b1(un,un-v)

Vu € W01,9(fi).

(4.17)

By (A3), we find that (Fiun - Fiu,un

- u)

= j £ ^ 1 [ ^ ( : r , u n , . D u n ) - A'i(x,u,Du)](un

-

Jn

= I Jn

u)Xidx

^^[A'^x^niDun)-Ai(x,un,Duj\{un-u)Xidx

+ / S ^ j [A'iix, un, Du) - A'i(x, u, Du)] (un Jn > / E ^ ! [ ^ ( a ; , w n , £ ) u ) - A'i(x,u,Du)](un Jn

-

u)Xidx u)Xidx.

Since u n —> u weakly in W0'9(Cl) and strongly in L9(f2), we easily see that E j l j [^(a;, un,Du)

- A[{x, u, Du)] -> 0 for a.e. a; e £1

Maximum

44

Principles

and

Applications

Therefore, for any given e > 0 there exists Qe C fi such that |fi £ | < e and HfLi [A'i(x, un, Du) - A'i(x, u, Du)] -> 0 uniformly on n \ Q€. By (A2), it follows that / E ^ [^-(z, u n , Du) - A'^x, u, Du)] (un Jn < f

u)Xidx

c0\un\q-l+c0\u\q-x+2c0\Du\q-l)\Dun-Du\dx

N{2k0 +

+ [ Jn\ne

\^1[A^(x,un,Du)-A!i(x,u,Du)]\\Dun-Du\dx

< Cdlfcoll^cn.) + llunlll;^.) + ll«lll; ( k) + \\Du\\l-lQc))\\Dun

-

Du\\LHa.)

+ | | E ^ ! [A'^x, un, Du) - Ai(x, u, Du)] || L ~(n\n e )\\Du n - £>«||z,i(n\ne) <

C I ( U M L ^ )

+ HunllKfn.) + HulllTfn.) + 1 1 ^ 1 1 ^ ) )

+ C 2 | | E i I 1 [A'^x, un, Du) - Afa, u, Du)] || L =o (n \n.) -> Ci(||fco|| L ,' ( n.) + 2ll«llL»(n.) + WDu\\r*(nc)) as r w oo. Therefore lim,»-.oo(-Fi««—-Fi".""—«) > —c'i(ll/i:ollJL«'(n«)+2llulliT(1ne) + lljDtxlli''(1n.))Letting e —> 0, we obtain liffln-^ooC^l""

_

FiU,Un - U) > 0.

On the other hand, (F\U, un — w) —> 0 as n —» oo. Therefore, by (4.16), limn_oo(-FiUn - Fiu, un - u) = lim n ^ 0 0 (i ? iu n , un - u) < 0. It follows that lim (Fiun - F\u, un-u)

— 0.

(4-18)

Due to (Al) — (A4), i<\ is of class (S)+, i.e., if u„ converges to u weakly in W0'q(tt) and (4.18) holds then un —* u in W01,9(fi) (see Lemma 3 in [Browder(1970)] or Theorem 2.1 in Chapter 1 of [Skrypnik(1994)]). Therefore from (4.18) we conclude that un —* u in W0'q(Q). By Lemmas 3.1 and 3.2 in Chapter 2 of [Ladyzhenskaya-Ural'tseva(1968)], we deduce that Tun —> Tu in W0'q(Q). Using this fact, we easily see that lim b?,(un,un — v) = b2(u,u — v) Vv € WQ'q(Q).

The Method of Upper and Lower

Solutions

45

Thus, in view of (4.17), we have b(u,u-v)

< l i m ^ ^ u , , , u n - v) Vw e W01,g(n).

This proves that F is pseudo-monotone. Finally we show that F is coercive. By (A4), we deduce Ejlj / A'i(x,u,Du)ux.dx

> a\\u\\q - Cx.

By (4.13) and our definition of p' we obtain | f Jo.

1

P'(x,u,Du)udx\
\\u\\Lqin).

From the definition of 7, we find that J n,(x,u)udx

> ( l / 2 ) H | * , ( n ) - C 4 ||U||.

By Young's inequality, for any e > 0, we have H ^ N I ^ n ) < (ei'/q')\\u\\i +

(e-yq)\\u\\lqm.

Therefore, if we choose e small enough and (3 > 0 large enough, we will have b{u,u) > (a/2)||u||« - C5\\u\\ -C0

Vu £

W^9(il).

This proves the coerciveness of F. We can now apply Theorem 4.8 with K = V = WQ'q(n) to conclude that the problem -A'u + p'{x,Tu,D{Tu))

+ /?7(x,u) = / in fi, w = 0 o n f f i

(4.19)

has a weak solution UQ € W0 'q(£l). Step 4. Let VQ = wo + g- Then w < vo < w and hence AVQ = Av0, Tvo = vo, 7(-, v0 — g) = 0, and vQ is a weak solution of (4.10). By definition, VQ satisfies E £ i / Ai{x,vo,Dv0)Xidx+ [ p(x,Tv0,D(Tv0))<j>dx Jn Jn +/? / 7 ( ^ ^ 0 - 5 ) ^ = / / ^ V^Wo1''^). in Vn

(4.20)

46

Maximum

Principles

and

Applications

Taking = (vo — w)+ we obtain / j(x,v0-g)4>dx=

/ [(v0 - w)+]qdx = \\(v0 ~

/ p(x,Tv0,D(Tv0))4> Jn

— / p(x, w,Dw)(v0 Jn

-

w)+\\lq{Q),

w)+dx,

and by (A3), T,^=1 / Jn

Ai(x,v0,Dv0)4>xidx Ai(x,w,Dv0)DXi(vQ-w)+dx

= EJIi / Jn

= S £ x / Ai{x,w,Dw)DXt(v0Jn +T,f=1 / LAi(x,W,DVQ) Jn

w)+dx — Ai(x,w,Dw)

> ££LX / Ai(x, w,Dw)DXi(v0 Jn

-

1

-

DXi(vo — w)+dx

+

w) dx.

Therefore from (4.20) we obtain / f(vo ~ w)+dx > Sf=! / ^4i(a;, w, Dw)DXi(vo — w)+dx Jn Jn + / p(x,w,Dw)(v0

- w)+dx + /3\\(v0 -

w)+\\"Lq{ny

Since w is a weak upper solution of (4.10) and (VQ — w)+ > 0, by definition, S^j

/ Aj(a;, to, Dw)DXi (v0 — w)+dx + / p(x,w,Dw)(v0 Jn Jn

— w)+dx

> / f(vo ~ w)+dx. Jn Thus we deduce ! f(vo - w) + dx > [ f(v0 - w)+dx + p\\(v0 - w)+\\L"(n)Jn Jn It follows that {3\\(v0 — w)+\\qLq,n) < 0 and hence (v0 — w)+ = 0, i.e., v0 < w. Similarly we can show that VQ > v. This proves Step 4 and hence finishes the proof of the Theorem. •

The Method of Upper and Lower

47

Solutions

Theorem 4.9 is due to Deuel and Hess [Deuel-Hess(1974)]. A natural question not addressed by this theorem is whether there exists a minimal and a maximal solution in the order interval [v,w] in W1,q(Q). Under a further condition on Ai, we can give a positive answer to this question by making use of the following result, which is essentially due to V.K. Le [Le(1998)] (see also [Dancer-Sweers(1989)] for a special case). L e m m a 4.10 that (A5)

Under conditions (Al) — (A4) for Ai, we suppose further Ai(x,t,£)

= Ai(x,£)

is independent

oft.

Let vi,V2 be weak lower solutions of (4-10); then m a x j v i , ^ } is a weak lower solution of (4-10). Similarly, if w\,W2 are weak upper solutions of (4-10), then so is min{u>i,u>2}Proof. We only prove the conclusion for weak lower solutions; the proof for weak upper solutions is analogous. Let v\,v2 S Wl'q(Q) be weak lower solutions of (4.10), and define v — max{ui,t>2}. We want to show that v is a weak lower solution of (4.10). We may write v = v\ + (u2 — ^ i ) + and by Theorem A.8, we find that

Dv1

in fti = { l £ [ l Vi > V2}

Dv = { Dvi Dvi = Dv Dv22

in ft<j ft0 •'= = {x {x £e ft •' vi Vi = — vV22}}

Dv2

in ft2 = { i e ( l V2 > V\ }

(4.21)

Therefore p(-,v,Dv)

=p{-,vi,Dvi)xsii

+p(-,v2,Dv2)xn\n1

€-L 9 '(ft).

Next we show that v < g on 5ft. To this end, we prove and use the following conclusion: If a,f3 e W^q{Q,), then max{a,/3} 6 WQ'"{Q.). Since (3 — a. £ WQ'q(fl), by definition, there exists rjn € Co°(fi) such that rjn —> j3 - a in the W 1,9 (ft) norm. It follows that, r)+ —> (/3 — a)+ in

Wx'q(ty.

It is clear that 77+ has compact support in ft, and therefore r)£ £ W£'9(n). Since w£'q{Sl) is a Banach space, we deduce (/3-a)+ e W£'q(Q) and hence max{a,(3} = a + ((3 — a)+ € W0,9(ft). Now we take a — (vi — g)+ and (3 = (V2 — g)+ and find that m a x ^ ! - g)+, (v2 - g)+} G

W^O).

Maximum

48

Principles

and

Applications

On the other hand, it is direct to check that max{(wi — g)+(x), (v2 — g)+(x)} = (v — g)+(x)

for a.e. i £ f i .

Hence (v - g)+ e W01,9(ft), i.e., v < g on dfl. It remains to show that a(v, )+/ p(x, v, Dv)dx < (/, ) V^ e W01,9(fi) with <> / > 0 a.e. in fi. By classical density results for Wl,q{0) (see Theorem A.11), there exists a sequence {wn} C C°°(f2) such that w„ ^ w := V2 — vi in W /1 ' 9 (fi). It follows that wn(x) —-> w(x) for a.e. x G fi. By passing to a subsequence, we may assume that

I K - H I <™_2>

V

(4-22)

«-

Let 7 : R1 —> i? 1 be a function such that (i) 7 e C - ^ 1 ) , (ii) 7 is nondecreasing in R1, (hi) 0 < 7 ( s ) < 1, (iv) 7(s) = 0 for s < 0, 7(s) = 1 for s > 1. Define Jn{s) = ^(ns). Then clearly j

n

satisfies (i)-(iii) above and

7„(s) = 0 for 5 < 0; 7„(s) = 1 for s > l/n. Moreover, let M — max{7'(s) : s G [0,1]}; then 0 < 7 ^(s) = nf'{ns)

< Mn.

(4.23)

Now for any <j> G Co°(f2), we define Tpl = V"" = (1 - In ° Wn)([>, 1p2=^2

= (in ° Wn)4>.

Clearly ipi,^ S Co°(^) a n d both functions are nonnegative. Since Vi, i = 1,2, are weak lower solutions to (4.10), by definition we have a(vi, ipi) + / p(x, Vi, Dvi)4>idx < (/, ip,), i = 1, 2. Jo. It follows that / ^LiAi{x,Dv{) -in{wn)(wn)Xi(j)-\-[\-^n{wn)\4)Xi dx L Jn + / p(x,vi,Dvi)[l-'yn(wri)]dx<(f,[l-'yn(wn)}),

(4.24)

The Method of Upper and Lower

Solutions

49

and / T,^L1Ai(x,Dv2)

in{wn){wn)Xi4>

+

^n{wn)^Xi dx (4.25)

+ / P{x,V2,Dv2)"/n(wn)(f)dx

< (/, 7n(™„ )(/>)•

Ja For a.e. x £ 0,2, w{x) > 0 and hence 7 n (u> n (x)) —> 1 for a.e. x £ Q2. Similarly, jn(wn(x)) —• 0 for a.e. x £ Qi. Adding (4.24) and (4.25), we obtain (/,<£)> / S j l i Ai(x,Dv2)

- Ai{x,Dvi)

in{wn){wn)Xi(f)dx

+ / S ^ ! Ai(x, Dv2) - Ai(x, Dvi)

jn(wn)cf}Xidx (4.26)

+

"E?=1Ai(x,Dvi)4>Xi

+

p(x,v2,Dv2)

-p(x,vi,Dvi)

Since p(x, V2,Dv2) = p(x, v\,Dv\)

(v /

+ /

Jni

Jn2'

)L

jn(wn)(pdx.

for a.e. x £ QQ, we have

/ p(x,v2,Dv2)-p(x,vl,Dvi) Jn L =

+p(x,vi,Dvi)(p\dx

J

-yn(wn)cf>dx

p(x,v2,Dv2)-p(x,vi,Dvi) ^n{wn)cj)dx.

By the dominated convergence theorem, we find, as n —> oo, / p(x,v2,Dv2)-p(x,vi,Dvi) Jn2 -> /

J

jn(wn)4>dx

[p{x, v2, Dv2) - p(x, vi, Dui)J 4>dx,

and / ./ni

p(x,v2,Dv2)

L

-p(x,vi,Dvi)

J

^n(wn)4)dx —> 0

Therefore, as n —> oo, / p(x,v2,Dv2) Jn L -> /

-p(x,vi,Dvi)

dx j

|p(x,w2,i?V2) -p(x,vi,Dvi)J^>dx.

(4.27)

Maximum

50

Principles

and

Applications

Similarly, as n —> oo,

hi

Ai(x, Dv2) - Ai(x,Dvi)

"/n(wn)cf>Xidx

JQ, 'i=l

^ Jf2i Jn2 j T,f=l Ai{x,Dv2) JQ.1

Ai(x,Dv2)

- Ai(x,Dvi)

- Ai(x,Dvi)

L

J

^n(wn)^Xidx

(4.28)

4>Xidx.

Next we use (AS), 7^(u>„) > 0 and > 0 to obtain

hi Jn

v4i(a;,Dv2) -Ai(a;,Dui) 7^(w n )(u 2

-vx)Xi4>dx

JQ „

(4.29)

+ / E ^

r

^ ( x , Dv 2 ) - Ai(x,Dvi)

r

~f'n{wn)(wn - w)Xi4>dx

i

> - / E-li A(a;,Dv2) -Ai(x,Dvi) L Jn

J

^'n{wn)(wn

- w)Xi<j>dx

By (4.22) and (4.23) we deduce that \\(wn - w)Xil'n(wn)4>\\Lq{n)

-> 0.

It follows that Jn

Ai(x,Dv2)

- Ai(x,Dvi)

i'n(wn)(wn

- w)Xidx

0.

Therefore, from (4.29) we obtain Iim„_ 0 0 / E j l i LU ( a ; , Z M ) - ^ ( x ^ i J) ^ ^ ) ^ ) * , ^ > 0. (4.30) Jn We now let n -> oo in (4.26) and make use of (4.27), (4.28) and (4.30) to obtain (/,<« >-\

Ef=1

Ai(x,Dv2)-Mx,Dvi)

f kl.Mx, Dv )cf> 1

Jn

+p(x,vi,Dvi)<j> dx

L

p(x,v2,Dv2) Jn2 — j T,i=1Ai(x,Dv)^)Xidx Jn

+/

Xi

,.dx

-p(x,vi,Dvi) + / p(x,v,Dv)dx. Jn

The Method of Upper and Lower

Solutions

51

Since Ai(-,Dv),p(-,v,Dv) G Lq (fi), we find that the above inequality also holds for arbitrary G W^'g(fl) with > 0 provided that any such is the limit in W 0 M (ft) of a sequence of nonnegative CQ° (fi) functions. Since CS°(Q) is dense in Wo'^fi), for any <j> G W01,9(fi) with 4> > 0, we can find 4>n G Co°(^) such that 0„ —» <> / in the W1+= <> / in W 1 , 9 (fi). Since cf>+ has compact support, we can use standard mollifier techniques as in Section 7.2 of [Gilbarg-Trudinger] to obtain a nonnegative function {+)tn £ Cg°((l) such that {cf>t)en -*m the W 1 ' ^ ) norm. This finishes our proof. • Theorem 4.11 In Theorem 4-9, if further Ai satisfies (A5), then (4-10) has a minimal weak solution u* and a maximal weak solution u* in the order interval [v, w] such that any weak solution u of (4-10) in [v, w] satisfies u* ) be as denned in Step 2 of the proof of Theorem 4.9. Then for any u G S, we have b(u-g,) = (f,tf>), V ^ e ^ ' ( n ) . By the proof of Step 3 there, b(u,u) > (a/2)||u||« - CslMI, Vu G W^q(tt). Therefore, for any u G S, (a/2)||u - S ||« - C 6 ||« - g\\
= (/, u - g) < C6\\u - g\\.

This implies that 5 is a bounded set in Wl>q(Sl). If {ui}i£i is a totally ordered set in S, we will show that u(x) := s\xpi€lUi(x) and u{x) = infjejUj(x) are elements in S. We only prove for u; the other case is similar. The conclusion is clear if I is finite. So we assume that / is an infinite set. Then we can find a sequence {un} in this family such that v < «i < U2 < ••• < w; lim un(x) = u(x). n—>oo 19

Since S is bounded in VF ' (r2), un must converge weakly to u in W1,q(Q). Since each un is a weak solution of (4.10), it then follows easily that u is a weak solution of (4.10). By Zorn's lemma, we conclude that (4.10) has a maximal solution u* and a minimal solution u* in [v, w] in the sense of

Maximum

52

Principles

and

Applications

partial ordering. We now use Lemma 4.10 to show that any weak solution of (4.10) satisfies u» u* and wi ^ u*. By Lemma 4.10, u\ is a weak lower solution of (4.10). Since u\ < w, by Theorem 4.9, there is a weak solution U2 of (4.10) such that ui < u u for all u £ S. Similarly, we can show that «» < u for all u € S. • The results in Theorem 4.9, Lemma 4.10 and Theorem 4.11 can be easily extended to Neumann boundary value problems. Let v = (V-\.,...,VN) denote the outward unit normal of dQ. and suppose that Ai satisfies (Al) — (A4). We say that u is a weak solution of -Au + p(x, u, Du) = f in ft, E£LjAi(x, u, Du)^ = 0 on dtt, if u e W 1 , 9 (n), p(x,u(x),Du(x)) a(u, v)+

belongs to L«'(ft) and

[ p{x, u, Du)vdx = (/, v) Vu <=

Whq(n).

A function u is called a weak upper solution of (4.31) if u € p(x,u(x),Du(x)) belongs to Lq (Q) and a(u, v)+

Jn

(4.31)

p{x, u, Du)vdx > (/, v) Vu G W^q(n),

W1,q(il),

v>0.

It is called a weak lower solution if the above inequality is reversed. If the imbedding W 1,9 (fi) t-» Lq(Q) is compact, one easily checks that the proofs for Theorem 4.9, Lemma 4.10 and Theorem 4.11 carry over with only minor natural changes when (4.10) is replaced by (4.31). Therefore we have the following result. T h e o r e m 4.12 The results in Theorem J^.9, Lemma 4-10 and Theorem 4-11 remain true if (4-10) is replaced by (4-31) and if the imbedding W 'q(Q.) ^-> Lq(Q) is compact (which is guaranteed if dQ, is Lipschitz continuous, see Remark A. 18). The above results can be extended to more general boundary value problems by making use of proper variational inequalities. Suppose that Ai satisfies (Al) — (AA). For convenience of notation, we write A(x,u,Du) = (AI(X,U,DU),...,AN(X,U,DU)). The other notations used above will be kept. Let K be a nonempty closed and convex subset of Wrl'<3'(Q), and

The Method of Upper and Lower

Solutions

53

(3(x,t) a function measurable in x for fixed t G R1, continuous in t for a.e. i £ ( l . We consider the following variational inequality on K: u G K : a(u,v

— u) + / p(x,u,Du)(v — u)dx Ja , + / j3(x, u)(v - u)dx > (/, v - u) \/v G K. Jdn

(4.32)

For the integral over dQ to make sense, we assume t h a t dQ is C 1 (or piecewisely C 1 ) so t h a t the trace of any u £ W1,q(tt) over dCl is well-defined and Tu = u\dn & Lr{dQ) for any r € [q, q*], where q* = (N - l)q/(N - q)\ when q < N, and 9* = 00 when q> N; moreover, T : W1,q(£l) —> L r (3J7) is compact if r E [<7, : $ - g G ^ ' ' ( f i ) } , t h e n it is easily seen t h a t u solves (4.10) if and only if u is a solution to (4.32). If we take K = Wl'q{Q), then (4.32) reduces to the variational equality u G Wl'q(Q)

: a(u,v)+ / p(x,u,Du)(v)dx , Jn + / p{x, u){v)dx = (/, v) Vv G

(4.33) W^iQ),

which is the weak form of the boundary value problem

{

— Au + p{x, u, Du) = /

in Q,

A(x,u,Du) • v = —(3(x,u) on dCl. This reduces to (4.31) if f3 = 0, and it becomes a Robin b o u n d a r y problem if P(x, u) = au. More generally, (4.32) can include mixed b o u n d a r y conditions and free boundary problems by suitable choices of K and /?; see [Le-Schmitt(2005)] for more examples. For functions u(x) and v{x) in W1,q{£i), we will use the s t a n d a r d notation (u A v)(x) = min{u(a;), v(x)}, and if V is a subset of W1'q(Q,), uAV

= {uAv:v£

{u V v)(x) = m a x { w ( i ) , u ( i ) } ,

we denote V}, uVV

=

{uVv:v€V}.

Following [Le-Schmitt(2005)], we have the following definition.

Maximum

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and

Applications

Definition 4.13 A function u G W1,q{Sl) is called a lower solution of (4.32) if the following conditions are satisfied: (i) p(-,u,Du) G Lq'(Q), /?(-,«) G L"°(n) for some g0 > q*/{q* - 1), where we understand q*\/{q* — 1) = 1 when q* = oo; (ii) uVK CK; (iii) for every v £u A K, a(u,v — u) + I p(x,u,Du)(v — u)dx Ja r + / (3(x, u)(v - u)dx > (/, v — u). Jan

(4.34)

Definition 4.14 A function u G W1,q(Sl) is called an upper solution of (4.32) if the following conditions are satisfied: (i) p(-,u,Du) G L«'(n), /?(-,u) G L*(fi) for some q0 > q*/(q* - 1), where we understand q*/(q* — 1) = 1 when q* = oo; (ii) u A K c K; (iii) for every v £ uV K, (4.34) holds. To see that the above definitions extend our earlier notion of weak upper and lower solutions for (4.10) and for (4.31), we first consider the case that K = W1,q(Q) and (5 = 0 in the above definitions. In such a case condition (ii) in Definitions 4.13 and 4.14 are trivially satisfied. Since {v-u:veu/\

Wlt9(Q)} = {-(u - v)+ : v G W1<9(Q)}

we find that (4.34) is equivalent to a{u,v) + / p(x, u, Du)vdx < (f,v) Vv G Whq{n),

v>0.

Therefore in this case, Definition 4.13 agrees with the notion of weak lower solution for (4.31). Similarly we can show that Definition 4.14 gives a weak upper solution for (4.31) (when K = Wl
-g£W^q(Q),

Vv&K,

The Method of Upper and Lower

Solutions

55

which is equivalent to (u — v)+ G WQ ,9(ft) for all v G K; i.e., U\QQ < v\dQ = g\aa, namely u < g o n dft. Moreover, using u A K = {u - (u - v)+ : v E K}, we find { u - u : t ; G u A i i ' } = {—(u - v)+ : v e K} = {-(u -g + w)+ :w£

W£'9(0)}.

Since (u — g)+ E ^ ' ' ( f t ) , it can be shown that { — (u — g + w)+ : w E Wo'q(fl)} is dense in {w < 0 : w G W01,
< (f,v) Vu € W01,9(ft), v > 0.

Thus in this special case, Definition 4.13 agrees with the notion of weak lower solution for (4.10). Similarly we can show that Definition 4.14 agrees with the notion of weak upper solution for (4.10). Let us now prove the following conclusion used above: If («o) + G Wo'^fi), then {(u0 + u)+ : u € W£'q(Sl)} is dense in {v > 0 : v G W 0 1,g (ft)}. For any given v G W 0 ' 9 (ft) with u > 0, we can find a sequence n G C£°(ft) such that u in W ^ f t ) . Hence <£+ -> w+ = v in ^ - ' ( f t ) . We note that + is Lipschitz continuous and has compact support in ft, say ft„. We now choose a C 1 function an such that 0 < an < 1, a n = 1 in a small neighborhood of ftn and an = 0 near 9ft. Then define , s

J4>t(x) -u0(x),

x G ft„,

[ - a „ u o - (1 - «n)«o i

zGft\ftn-

It is easily checked that un G WQ,q(fl) and (MQ + w«) + = 4>n- This proves our conclusion. Theorem 4.15 Let (Al) — (A4) hold and Q be a bounded domain with piece-wisely C 1 boundary dfl. Suppose that v and w are lower and upper solutions of (4-32), respectively, and v < w a.e. in ft. Suppose further that there exist a constant C\ > 0 and functions k\ G Lq (ft), ki G V (9ft) such that \P(x,t,0}
\P(y,s)\
(4.35)

56

Maximum

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and

Applications

for a.e. x G CI and y G dCl, all £ G R and all t G [v(x),w(x)], s G \v(y),w(y)], where q' = g/(g - 1), r' = r/(r — 1) and r G [q,q*)- Then (4-32) has a solution u satisfying v < u < w a.e. in CI. Proof. We follow the arguments in the proof of Theorem 4.9. Let Ai(x,t,£) and T be defined as in Step 1 of the proof of Theorem 4.9. Let j(x,t) be defined as in Step 2 there but with 5 = 0. We now define b(u, 4>) = bi (u, (j>) + b2(u, 4>) with h{u,4>) = ^fLi

/

Ai(x,u,Du)4>Xidx,

JQ.

b2{u,)= I p(x,Tu,D(Tu))4>dx Ju

+ k I ^{Xj^^dx JQ

+ / (3(y,Tu)4>dy. Jdn

Then there exists a unique F : Wl'q(Cl) -> (Wl>q(Cl))* such that b(u,(f>) = (Fu,4>) V £

W1'q(fl).

We want to show that F is bounded, pseudo-monotone and coercive. The boundedness of F is easy to see. As in the proof of Theorem 4.9, there exists a pseudo-monotone operator Fi : W1,q(Cl) —> (W1,q(fl))* such that bi(u,) = (FlU,(t>) V0 G

W1'"^).

Moreover, by (A3), if un —> u weakly in W1'q(Q) and lim„^ 00 (FM„,M n — u) < 0, then we can prove as in Step 3 of the proof of Theorem 4.9 that t i „ - m strongly in W1,q(Cl). It follows that lim b2(un,un — v) = b2(u,u — v) Vv G W1,q(Ci). n—*oo

Since we already know that F\ is pseudo-monotone, we have b(u, u-v)<

lim n .^b(u„, un - v) Vv G W 1 ' 9 ^ ) .

This proves that F is pseudo-monotone. As in the proof of Step 3 in Theorem 4.9, we easily obtain ££LX / Ai(x,u,Du)uXidx in | [ Ja

> a\\u\\q — Ci\\u\\,

p(x,Tu,D{Tu))udx\-1\\u\\Lq{a),

The Method of Upper and Lower

Solutions

57

and

f

j(x,u)udx > \\u\\lq(a) - C4\\u\\. la Ja For the boundary term, we have

'I.

P{y,Tu)udy\

an

< \\k2\\L^(m)\\u\\Lr(dQ)

< C5\\u\\.

Therefore, using Young's inequality as before and taking the constant k sufficiently large, we deduce b{u,u) > (a/2)||«||« - C 6 ||u|| Vu €

W1'"^).

This proves that F is coercive in W1,q(Q). It follows that (4.11) holds with A = F and any VQ £ K provided that p is large enough. Therefore, there exists a solution for the variational inequality u G K : (Fu, 4>-u)>{f,(j)-u)

\/€ K.

(4.36)

If we can show that v < u < w for any solution u e K of (4.36), then u also solves (4.32) and the conclusion in our theorem is proved. To check that u < w is satisfied, we consider the function <j) = w f\u = u — (u — w) +. Since u £ K,we find £ K by our assumption that w is an upper solution. Using this (f> in (4.36), and notice that / (3(y,Tu){u-w)+dy= Jan

(3(y,w)(u -

w)+dy,

Jan

we obtain, as in Step 4 of the proof of Theorem 4.9, that (f,{u-w) +

)>b(u,(u-w)+) > a(w, (u — w)+) + / p(x, u>, Dw)(u — w)+dx

On the other hand, since w is an upper solution to (4.32), we have a(w, ip — w) + / p(x,w,Dw)(ip Jn +

— w)dx

/3(y,w)(ilj-w)dy>(f,ip-w)ViJj£wVK. Jan

Maximum

58

Principles

and

Applications

Taking tp = w\/ u = w + (u — w)+, we deduce (/, (u — w)+) < a(w, (u — w)+) + / p(x, w, Dw)(u — w)+dx

Jn + /

f3(y,w)(u-w)+dy.

JdQ.

It follows that (/, (« - «/)+) > k\\(u - w ) + | | [ , ( n ) + (/, (u - w)+). Therefore u < w a.e. in fi. Similarly we can show that u > v. This completes the proof. • Remark 4.16 (i) The condition for p(x, t, ^) in (4.35) can be slightly relaxed; Theorem 4.15 still holds if this condition is replaced by

IpOM.OI^MaO+CiKI'"1, where fci G L«'(ft), 2 Au>2Proof. Suppose that v\, V2 are lower solutions. Then v\ V K C K, V2 V K C K. It is easy to check that (tiVo)Vt« = uV(i)Vt»). Therefore (vi V v2) V K = vi V (v2 V K) c vi V K c K. Similarly, if ui\, w2 are upper solutions, then (u>i A u>2) A K C K. To check that vi VV2 satisfies (i) and (ii) in Definition 4.13, we can follow the proof of Lemma 4.10 with only minor changes. Therefore vi V v2 is a lower solution to (4.32). Similarly, u>\ A u>2 is an upper solution to (4.32). • With the above lemma, we can repeat the proof of Theorem 4.11 to obtain the following extension.

The Method of Upper and Lower

Solutions

59

Theorem 4.18 In Theorem 4-15, if further Ai satisfies (A5), then (4-32) has a minimal solution u„ and a maximal solution u* in the order interval [v,w] in K such that any solution u of (4-32) in [v,w] satisfies u* < u < u*. Remark 4.19 It is unknown whether condition (A5) is necessary for Lemma 4.17 and Theorem 4.18 to hold.

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Chapter 5

The Logistic Equation

The simple looking logistic equation is a basic model in many reactiondiffusion problems. In this chapter, we first consider the classical case, which can be satisfactorily understood by making use of the principal eigenvalue and upper and lower solution method. In Section 5.2 we consider the degenerate case, which is technically more demanding and induces a boundary blow-up problem (which will be studied in detail in Chapter 6). It turns out that, in the degenerate case, the equation can behave very differently from the classical case. Section 5.3 uses a perturbation approach to understand the change of behaviour of the logistic equation as it gradually changes from the classical case to the degenerate case, and it reveals interesting qualitative properties of the solutions.

5.1

T h e classical case

Consider the following problem —Au = Xu — b(x)up in fi, « = 0 o n dQ,

(5.1)

where flis a bounded domain in RN with C 2 boundary dQ, b(x) is a positive continuous function over Q and p > 1 is a constant; X e R1 will be regarded as a parameter. Problem (5.1) is some times called a logistic equation, which can be used to describe the steady-state population distribution of a species whose growth obeys a diffusive logistic law. It is a basic model and plays important roles in the study of many reaction diffusion problems. We are interested to know when (5.1) has a positive solution. Note that by the strong maximum principle, or more generally, Theorem 3.9, we know that any nonnegative solution of (5.1) is strictly positive in Q unless it is identically 0. By a positive solution, we mean a solution u G W2'q(Q) 61

62

Maximum

Principles

and

Applications

(q > 1) which is strictly positive in Cl. Theorem 5.1 Let Ai denote the principal eigenvalue given in Theorem 2.7 with L = A. Then (5.1) has no positive solution when A < Ai and it has a unique positive solution if A > Ai. Proof.

Let <j> > 0 be such that —A = A ^ in Cl, <j) = 0 on dCl.

Suppose that (5.1) has a positive solution u. Then multiplying (5.1) by <j>, integrating over Cl and using integration by parts, we deduce Ai / cj)udx = A / (fiudx — / b{x)up4>dx < A / cfiudx. JQ Jn Ja. Ja

Therefore A > Ai. This implies that (5.1) has no positive solution when A < Ai. Suppose now A > Ai. It is easily seen that for all small e > 0, -A(e) = \!{e<j>) < X(ecf>) - b{x)(ecf>)P in Cl, ecj>\9a = 0.

Therefore e<j> is a lower solution of (5.1). On the other hand, it is clear that any positive constant M satisfying M p _ 1 min^fr > A is an upper solution. Hence for small e > 0 and large M > 0, v := ecf> and w := M form a pair of lower and upper solutions of (5.1) with v < w. We can now apply either Theorem 4.5 or Theorem 4.9 (and standard Lp regularity) to conclude that (5.1) has a solution u € W2,q(CT) (\/q > 1) satisfying v Ai. By the Hopf boundary lemma (see Lemma A.35), we have Dv4> < 0 and DvUi < 0 on dfl for i = 1,2, where v denotes the outer unit normal of dfl. This implies that for sufficiently small e > 0, v = e < ut in Cl for i = 1,2. Therefore, if w = M is sufficiently large, the minimal solution u* of (5.1) in [v, w], whose existence is guaranteed by Theorem 4.5 (or Theorem 4.11), must satisfy M* < ut, i = 1,2. We now multiply ut to the equation satisfied by u„, integrate by parts over fl and obtain f 6(a;)u,u i («r 1 - u^dx

= 0.

Since w* < Mi, this is possible only if u, = w*, contradicting our assumption that u\ and ui are different solutions. •

The Logistic

Equation

63

In the above proof, the self-adjointness of A was used in the integration by parts process. This can be avoided by making use of the properties of the principal eigenvalue. Let Xi(tp) denote the principal eigenvalue of the operator A — ip, where tp is a function in i°°(fi). Lemma 5.2

The following conclusions hold true:

(i) Ifipi < tp2, then (ii) Ifipn-^ip

AI(T/>I)

< Xi(ip2) and equality holds only z/i/'i =

in L°°{tt), then Xi{ipn) - • Ai(V>).

Proof. For part (i), we use Theorem 2.8 and find that there exists 4>i > 0 such that A<^i ~ipii = 0 in fl, (j>1\dQ = 0. Therefore A0i - 4>2i + Xi(^i)4>i < 0 in fl, cj>i\dn = 0. It follows from Theorem 2.8 again that the principle eigenvalue of the operator A—02+Ai(V'i) is positive unless ip2 = tyx- On the other hand, it is easy to check that the principal eigenvalue of this operator is X\{^>2) — Ai(V'i)This proves part (i). To prove part (ii), we denote by cf>n and <j> the normalized positive eigenfunction corresponding to X\(ijjn) and Ai(V>)> respectively: -A<j>„ = -i>nn + Xi(tpn)(j)n in il, 4>„\dn = 0, \\<j>n\\oo - 1,

(5.2)

-A = -i/f + \i(il>) in n, 4>}dQ = 0, I^H^ = 1. Since i/jn —> , we may assume that ip — 1 < ipn + 1 for all n > 1 and therefore, by part (i), A i ( V ' - l ) < A 1 ( V ' „ ) < A i ( ^ + l). It follows that —ipn(f>n + Ai(V>«)0n has a bound in L°°(fi) independent of n. Therefore, {0„} is a bounded set in W2'q(Q) for any q > 1. By Sobolev imbedding theorems, {^n} is precompact in C x (f2). By passing to a subsequence we may assume that n —> o in C 1 ^ ) . We may also assume that Xi('n + Ai(V>„)(/)„ -* -V^o + X*(j)0 in L°°(n)

64

Maximum

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and

Applications

and hence, by (5.2), we deduce that n —> o in W2,q(£l) for any q > 1, and -A(/>o = -V"^o + A > 0 in fi, <^0|ai7 = 0, ||Q > 0. By the strong maximum principle, we deduce o > 0 in A. Therefore A* must be the principal eigenvalue for A — ip, i.e., A* = Ai(-0), and (j>o must be the uniquely determined normalized positive principal eigenfunction <j>. It follows that the entire original sequence „ converges to <j> and \i{ipn) —> Ai(?/>). • Let us now see how Lemma 5.2 can be used to replace the integration by parts process in the proof of Theorem 5.1. Suppose that (5.1) has a positive solution u. Then we rewrite (5.1) in the form Au — [6(x)wp_1]u + Aw = 0 in Cl, U\QQ = 0, and deduce from Theorem 2.7 that A = Ai(6u p _ 1 ). Using part (i) of Lemma 5.2, we obtain A > Ai(0) = Ai. Similarly, from u» < uj, u* < UQ, and equality does not hold for both, we deduce

Ai(K_1) < MK" 1 ). A i( 6 < _1 ) ^ M K - 1 ) , and strict inequality holds at least in one of the above. But on the other hand, A 1 ( K - 1 ) = A 1 (6 w f- 1 ) = A. This contradiction proves the uniqueness part of Theorem 5.1.

5.2

The degenerate logistic equation

When N > 3 and p = ^ f , equation (5.1) is closely related to certain problems in geometry. In a geometric setting, Kazdan and Warner [KazdanWarner(1975)] asked whether the conclusion of Theorem 5.1 remains true if one only assumes b > 0, ^ 0. The case that b(x) has a nontrivial vanishing set is also useful in capturing the effects of heterogeneous environment on various population models, some of which will be discussed in volume 2. Problem (5.1) with such a b(x) is known as the degenerate logistic equation.

The Logistic

Equation

65

It turns out that the Kazdan-Warner question has a negative answer (see [Ouyang(1992)]). The following result is a special case of Theorem 2.1 in [Du-Guo(2006b)]. Theorem 5.3 Suppose that b(x) is continuous, nonnegative and not identically 0 in ft, and ftg := {x G ft : b(x) = 0} has nonempty interior fto. Then there exists X* G (A^Aj 0 ] such that (5.1) has a unique positive solution u\ for X £ (Ai\A*), and no positive solution otherwise. Moreover, l|uA||L~(n) - » o o as A increases to A*. Here we use XP to denote the principal eigenvalue of A over D under 0 Dirichlet boundary conditions; X1 ° is understood in the sense of [BerestyckiNirenberg-Varadhan (1994) ]. Proof. Let ip > 0 be the unique L°°(ft)-nor:malized eigenfunction corresponding to Aj\ and define ak = X^{kb(x)^-1(x)),

fc

= l,2,...

By Lemma 5.2, we deduce Xr < a\ < a 2 < ... < ak < ak+\ < ... Using Theorem 2.8, it is easy to see that if ft' is a proper subdomain of ft (i.e., ft' c ft and ft' jt ft), then A?(<0) < A ? ' ( $ for any 4> e L°°(ft). Therefore ak<X^°(kb(x)^-1(x))

= X20.

This implies that A* := lim ak < A^°. k—>ao

We now prove that (5.1) has a positive solution for each A £ (A^2, A*). Since ak increases to A*, there exists k such that X < ak. Let k > 0 denote the unique normalized eigenfunction corresponding toojt: -A(j>k

= -kb(x)i()p~1(x)(t)k

+ ak(f>k, fc||oo = 1-

Then due to Hopf's boundary lemma, there exist positive constants c\ < ck < C2ip in Q. Therefore we can find a constant M —

66

Maximum

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Applications

M(k) > 0 sufficiently large such that A(M0 fc ) + \{M4>k) - b(x)(Mk)p + \{Mcj>k) - b(x)(Mcf>k)p

= -ak(Mk) + kr-\x)b{x)(M4>k) = (A - ak)(M<j>k) + Mb(x)^k

[k^-1

- (M^)*"1]

< 0 for all a: e ft. This implies that Mk is an upper solution of (5.1). On the other hand, for e > 0 sufficiently small, we can easily check that tip is a lower solution of (5.1) with eip < M 0 in ft satisfying -AV>* = a > * - b(x){ip*)p in

ft,

ip* = 0 on dft.

It follows from this equation that a* = A? W T

_ 1

) > A?.

As a* g (Ap, A*), we must have a* > A*. On the other hand, due to the Hopf boundary lemma, [ip*(x)]p~1 T _1 ) < Xfik^^b)

=ak<

A*.

This is in contradiction with a* > A* and hence proves what we wanted. Finally, we show that ||UA||L~ —> oo as A | A*. If this is not true, then, we can find a sequence {an} satisfying an | A* as n —• oo such that {||"a„IU~} is bounded. It then follows from the Lp theory and Sobolev imbedding theorems that there exists u £ C^ft) such that, by passing to a subsequence, un —> u in C 1 (fi) n W2 0. We claim that u > 0. Indeed, since a n increases with n, u a„ + 1 is an upper solution for the equation satisfied by uUn. As before, for small e > 0, eip is a lower solution and hence there exists a solution u in [«/»,M 0 „ + I] for (5.1) with A = an. Since we already know that uan is the

The Logistic

Equation

67

unique positive solution, we must have u = uan and hence wa„ < M a„ +1 Therefore u > uan > 0. But this is in contradiction to the following fact proved above: (5.1) has no positive solution when A > A*. • R e m a r k 5.4 (i) The conclusions in Theorems 5.1 and 5.3 hold when A is replaced by a general second order elliptic operator, not necessarily self-adjoint. Moreover, the smoothness requirement for dQ can be considerably relaxed, (ii) It is easy to show that u\ —> 0 uniformly in Q as A deceases to Ap. (iii) The limit of at in the proof of Theorem 5.3 could be finite even if fio has empty interior; see [Dancer(1996)] for more details. The behavior of u\ near A — A^ is usually called bifurcation from 0: as A increases across Af, a branch of nontrivial solutions u\ bifurcates from the trivial solution u = 0. It can be further proved that ux/\\u\\\Loa^ —• ip uniformly on fi as A decreases to Af, where tp > 0 is the £°°(fi)-normalized eigenfunction corresponding to Ap. Therefore one may say that the bifurcation from 0 near A = A" is largely determined by the linear equation —Au = \^u in fi, u\an = 0. We next examine the point-wise behavior of u\(x) as A —> A* in more details. We already know by Theorem 5.3 that ||uA||L°°(n) -> oo as A increases to A*; such a phenomenon is usually called bifurcation from infinity at A = A*. To simplify our discussions to follow, we assume that fig = ^o C Cl and fio is connected with C 2 boundary d£lo-

(5.3)

We will see that to fully describe the behavior of u\ as A increases to A*, we will need a linear problem as well as the following boundary blow-up problem: —Ait = Xu — b{x)up in fi \ Qo, u\da0

=

°°i u\on = 0-

(5-4)

Here by u\d^0 = oo, we mean u(x) —> oo when d(x, dClo) —* 0. We now list some results for (5.4) required for our description of u\ while leaving a detailed discussion of this and other related boundary blow-up problems to the next chapter.

Maximum

68

Principles

and

Applications

Proposition 5.5 Suppose that the conditions in (5.3) are satisfied. Then, for any A G R1, problem (5.4) has at least one positive solution. Moreover, it has a maximal positive solution U\ and a minimal positive solution U_x in the sense that any other positive solution u satisfies U_\ 0 and /32 > P\ > 0 such that for all x € Q\CIQ near 8Q,Q,

A[d(x,dih)] a < b(x) <

p2[d(x,dn0)]a,

then (5.4) has a unique positive solution. The following comparison result will also be needed (it improves Lemma 2.1 in [Du-Ma(2001)]]). Lemma 5.6 Suppose that D is a bounded domain in RN, a{x) and f3(x) are continuous functions in D with \\a\\ Ltx ^ < oo, and /3(x) is nonnegative and not identically zero. Let u\,U2 £ C 1 (fi) be positive in H and satisfy in the weak sense Lu\ + a(x)ui - )3(x)g(ui) < 0 < Lu2 + a(x)u2 — /3(z)g(« 2 ), x e D, (5.5) and \imx^QD(u2 where Lu = T,ij[aij{x)uXi\Xj atj = ajU

C l |^|

2

- ui) < 0,

with aij G L°°(D)

< ZijOijixKiZj

satisfying

< c 2 |£| 2 Vz € D, £ € RN,

(5.6)

for some positive constants c\ and C2, and g(u) is continuous and such that g(u)/u is strictly increasing for u in the range infp{Mi,U2} / P(x)[g(u2)w2 - g(ui)wi]dx + / a(x)(uiw1 JD

JD

(5-7) - U2W2)dx.

JD

For e > 0, denote €\ = e and e2 = e/2 and let Vi = [(«2 + e 2 ) 2 - (ui + e i ) 2 ] + / K + e0, i = 1,2. Clearly Vi is Lipschitz continuous and vanishes near dD. Therefore, by standard techniques of mollifiers (see Section 7.2 in [Gilbarg-Trudinger]) Vi

The Logistic Equation

69

can be approximated arbitrarily closely in the Wl'2(D) n L°°{D) norm by C°° functions vanishing near dD. Hence (5.7) holds when Wi is replaced by Vi, i = 1,2. Denote

D + (e) = {x£ D: u2{x) + e2 > Ui(x) + e ^ .

We see immediately that the integrands in (5.7) (with Wi = v^ vanish outside D+(e). Moreover, using the simple observation that Ea^- (uXi vx. + vXiuXj) = ^aijUXivXj, one easily checks that the integral on the left hand side of (5.7) equals

If :=

- /,+w ~^°IJ

[{U2U

+Sa i j[(ui) a ; .

" 5T?{Ul)xi] [{U2h" Srt ( W l ) ^

1

:

("2)xi] [(«i)ij

1

1

(U2)xj]\dx,

which is non-positive. On the other hand, as e —> 0, the first term on the right hand side of (5.7) converges to

/

P(x)(g(u2)/u2

- g(ui)/ui)(ul

- «i),

JD+(O)

while the last term in (5.7) converges to 0. Therefore, we would have a contradiction unless D+(0) C {x £ D : (3(x) = 0}, that is, /3(x) = 0 whenever U2(x) > ux(x). Therefore, by (5.7),

iim^oie > 0.

If U2(x) > Ui(x) holds for some x, then -D+(0) is a nonempty open set, and for any open subset D 0 satisfying D 0 C D + (0), we have D+(e) D DQ for

70

Maximum Principles and Applications

all small e > 0, and hence, due to (5.6), lim^o-fe < lim e ^o / -\^aij[(u2)Xi L JDo I +T,aij[(ui)Xi

• (ui)Xi\ [("2)*,3 MI + e i ——-(u2)Xi] [(ui)Xj U2 + £2

= /

- { S a y [(u2)Xi

JDo

L

U

+Haij[(u1)Xi < -Cl(\Du2 V

U2 + £2

-{ui)Xi] [(u2)Xj l

+ |I>«i -

'

u

U

-( i)Xj] l

-(u2)Xi] [(ui)Xj

- ^-Dui\2 u\

; (ui)xA wi+ei —. (u2)Xj] \dx

~(u2)Xj]jdx

^-Du2\2)\D0\. u2 J

On the other hand, lim^o-fe > lim,. ,0L > 0. Therefore, \Du2 - —Dm] = \Dm - —Du2\ = 0 in D0. U\

U2

It follows that |r>(ui/u2)|=0in£>0. This implies that \D(ui/u2)\ = 0 in JD+(0) and hence ui/u2 is a constant over any component of D+(0). Let M\ be such a component. Since (3 = 0 in -D+(0) and (3 jk 0 in D, we necessarily have dM\ n Z) 7^ 0. For any x G <9Mi n D, ui(x) = u2(x) > 0 and hence ui(x)/u2(x) = 1. It follows that ui/u2 E 1 in Mi, contradicting the fact that u2 > ui in D+(0). This shows that D+(0) must be empty, i.e., u2 < «i in D. • We are now ready to give a rather complete description of u\ for A close to A*. Theorem 5.7 Suppose that the conditions in (5.3) are satisfied. A* = Ax ° and the following hold: (i) u\ —* 00 uniformly on Q.Q as A increases to A*; (ii) u\ —+ C/A» uniformly in any compact subset ofQ\flo toX*.

Then

as X increases

The Logistic Equation

71

Proof. We first observe that u\ increases as A increases. This follows from a simple upper and lower solution consideration, together with the uniqueness of u\; such an argument was used near the end of the proof of Theorem 5.3, so we omit the details. The monotonicity of u\ in A implies that if we can prove (i) and (ii) along a sequence An —» A*, then the same conclusions hold for A —> A*. Let {A„} be a sequence of positive numbers such that A^ < An < A* and An increases to A* as n —> oo. To simplify notations, we write un — u\n. The proof below is divided into 4 steps. Step 1.

A* = Aj ° and un(x) —> oo uniformly in any compact subset of

Denote un = u n /||u„||oo- Then -Aw„ = Xnun - &(:r)||un||£71u£ < A*u„ in Q,, un\ga = 0. It follows that f \Dun\2dx < X* [ u2ndx < X*\n\. Jn Jn Therefore {«„} is bounded in the Hilbert space Wo'2(Q.). This implies that, subject to a subsequence, un —> u weakly in W0' (fi) and strongly in L2(Q) for some u G WQ' (Q). Since {u„} is bounded in L°°(fi), its convergence to u in L2(f2) implies that it also converges to u in Lq(D,), for all q > 1. Clearly we must have u > 0. We claim that u jk 0, for otherwise, (—A) - 1 u„ —> 0 in L°°(Q.) by standard Lp regularity, and due to 0 < un < A * ( - A ) - 1 u n , we deduce un —> 0 in L°°{Q), which is a contradiction to the fact that ||Wn||oo = 1For any given G C£°(fl), we multiply ^/||u„||oo t o the equation satisfied by un, integrate over fl and apply integration by parts, and obtain / Dun • D(f>dx = A„ / un4>dx — Hwnll^"1 / b(x)u^cf>dx. (5.8) Jn Jn Jn Since ||wn||oo —> oo by Theorem 5.3, we divide (5.8) by Hunll^ 1 , let n —> oo, and obtain / b{x)up(j)dx = 0. Jn Since b is nonnegative and b(x) > 0 in ft \ Clo, and <j> is arbitrary, we must have M = 0 on fi \ fio- This implies that u|n 0 belongs to W0' (^o), and it is nonnegative but not identically 0 in QQ.

72

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and

Applications

On the other hand, if in the above discussion, we have chosen € C£°(fio), then (5.8) reduces to /

Dun • D(f>dx = An j

un4>dx.

/fio

Letting n —> oo we obtain / Therefore u\n0

ls

Du • D(j>dx = A* / ii(f>dx.

a nontrivial nonnegative weak solution of the problem -Au = X*u in flo, w|an0 = 0.

By Harnack's inequality we find u > 0 in Qo and by regularity bootstrapping, w|n0 £ C2(Qo)- By Theorem 2.7, we deduce A* = A 1 °, and U\Q0 is the unique positive normalized eigenfunction corresponding to X1°. This implies that the entire original sequence un converges to u in Lq(Q,). For any small S > 0, let £ls = {x £ fio : d(a;, <9fio) > <$}• Since —Aun = Anwn has a bound in L°°(fio) which is independent of n, it follows from the interior Lp estimates that {u„|n 4 } is bounded in W2,q(fls) for any q > 1. Therefore, subject to a subsequence, un\(is converges to U\Q5 in C 1 ^ ^ ) . Since u is uniquely determined, the entire sequence converges. This implies that for any compact subset K oi flo, un —> u uniformly in K. Since u is continuous and positive in Qo, and ||un||oo —* oo, we find that un —> oo uniformly in K. This proves Step 1. Since <9fi0 is C2, it satisfies a uniform interior ball condition: There exists R > 0 such that for any x 6 dflo, there is a ball Bx of radius R such that Bx C fio and Bx n c*f20 = {x}. S t e p 2. Le£ x n 6 Sf2o &e suc/i £/ia£ Un(xn) = min u„(x). xgSOo u

If { n{xn)}

is bounded, then we can find a constant a > 0 and a sequence

cn —> oo SMC/J t/iai

u n (x) > M „ ( I „ ) + c„V>(x), w/iere ^(x) = e _!7 l x_yTl l - e _<7fl A simple calculation gives

whenever R/2 < \x — yn\ < R, anc? ?/„ is £/ie center of the ball BXn.

AV> + AnV> - (4a2\x - 2/„|2 - 2Na + Xn)e-a\x-^a

- A„ e - CTfl2 .

(5.9)

The Logistic

Equation

73

We can choose a large a > 0 such that -AV>(z) < \nil>(x),

Vx £ BXrt \

BR/2(yn),

where BR/2(yn) = {x £ RN : \x - yn\ < R/2}. Choose a compact set K CC fio such that K D U%LlBR/2{y„). By Step 1 and the assumption that {un(xn)} is bounded, we can find a sequence c„ —> oo such that un(x) > un{xn) + cn(e-aR2/4

- e~°R2),

Vx £ BR/2(yn)

C K.

On the other hand, since An < A*, by the maximum principle, un(x) > Un(xn), Vx £ Cl0. In particular, un{x) > un{xn) on dBXn. Thus we see that un is an upper solution to the problem - A u = \nu in BXn \ BR/2{yn) < u\dBXn — un(xn), . u\dBR/2(yn) = un(xn) + cn{e-aR2lA But clearly, un(xn)

(5.10) aR2

-

e~ ).

+ cntp(x) is a lower solution to (5.10). Hence, since

B

A„ < A* < X[ ^\BRMV^))!

un{x) > un(xn)

by T h e o r e m

+ cnip(x),

2.8,

whenever R/2 < \x — yn\ < R,

as required. S t e p 3. limn-^oo un(x) = oo uniformly on CloBy the maximum principle, it suffices to show that Un(xn)

=

m

i

n

Un(x)

—> OO.

We argue indirectly. Suppose that this is not true. Then by passing to a subsequence, we may assume that {un(xn)} is bounded: un{xn) < C for all n. Clearly un is an upper solution to - A M = \nu - b*up in Q. \ fio; w|an0 = un(xn),

U\QQ = 0,

(5.11)

where b* = ||6||oo- Since 0 is a lower solution, we see that (5.11) has a positive solution vn < un. Replacing un(xn) in (5.11) by its upper bound C, we similarly obtain a positive solution V of (5.11) satisfying vn < V on fi\f2o- In particular, ||wn||.L°°(n\P.0) ^s bounded. Then the L p -estimates and

74

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Applications

the Sobolev imbedding theorems imply that {vn} is bounded in C 1 (f]\fio)In particular, \Dvn(xn)\ is bounded. Since un(x) > vn(x),

Vx G fi \ fio, and un(xn)

=

vn(xn),

we have dun{xn)/dvn

< dvn{xn)/dvn

for some Co > 0, where vn = (yn — xn)/\yn On the other hand, by Step 2, dun(xn)/dvn

> cndip{xn)/dvn

< C0

— xn\, and yn is as in Step 2.

— cn[2aRe~aR

] —> oo

as n —> oo. This contradiction finishes the proof of Step 3. Step 4. For any compact set K C fl\ flo, un —> f/A, in C 1 (K) as n —> oo. We can applying Lemma 5.6 to deduce that u„ < U_x. in fi\Qo- Since un is increasing, Moo^) : = limn->oo n„(i) exists for a; G fi\Ooj and WQQ < U_\»By the Lebesgue dominated convergence theorem, we find that un\K —> q UOO\K in L (K) for any g > 1 and any compact subset K oi Q.\Q,Q. It follows that UU\K —> Uoolic in W 2 ' 9 ^ ) for any g > 1. Therefore un —> UQQ inC1^). It is easily seen that Uoo satisfies (5.4) with A = A*; the fact that Moo = oo on dClo follows from un{x) < un+i(x) and un(x) —> oo uniformly on dQ,o by Step 3. Since U_x, is the minimal solution, we necessarily have u oo = IL\* • This proves Step 4 and hence finishes our proof of Theorem 5.7. • Remark 5.8 (i) Theorem 5.7 and its proof show that the behavior of u\ as A —» A* is determined both by the boundary blow-up problem (5.4) and by the linear eigenvalue problem associated with A^°. Note that from the proof in Step 1, UA/||"A||OO —> ipo in L2(Cl), where ^o denotes the normalized positive eigenfunction corresponding to Af °, extended to be 0 outside fl0. (ii) Further related results to Theorem 5.7 can be found in [DuHuang(1999)], [L6pez-G6mez(2000)], [Du-Guo(2003)] and [DuOuyang(2002)].

The Logistic

5.3

Equation

75

Perturbation and profile of solutions

Suppose that the conditions in (5.3) are satisfied. For small e > 0, we now consider the perturbed problem - A u = Au - [b(x) + e]up, u\dn = 0.

(5.12)

By Theorem 5.1 we know that (5.12) has a unique positive solution u\ when A > Xf, and there is no positive solution otherwise. This is in sharp contrast to the case e = 0 described by Theorems 5.3 and 5.7. We now examine how (5.12) evolves as e decreases to 0. Our results below show that (5.12) is not sensitive to the change of e as e —» 0 in the case A < A^°, but it is very sensitive to this change of e when A > A: °. This observation turns out to be useful in capturing the effects of heterogeneous environments on a variety of reaction diffusion models (see [Du(2004a)] and the references therein). Theorem 5.9 Suppose that the conditions in (5.3) are satisfied and let u\ and u\ be the unique positive solutions to (5.12) with e = 0 and e > 0, respectively. Then the following hold: (i) If Ap < A < A: °, then u\ —> u\ uniformly on Q as e —> 0. (ii) IfX>Xi°, then (a) u\ —> oo uniformly on Qo o,s e —> 0, (b) u\ —>• U_y uniformly on compact subsets o/fi \ fio as e —> 0. Here U_x is defined in Proposition 5.5. Proof. Recall that by an upper and lower solution consideration and the uniqueness of u\, we deduced that u\(x) is increasing in A. The same consideration can be used to show that u\(x) is increasing in A, decreasing in e, and u\{x) < ux(x)

(5.13)

whenever both exist. (One can also apply Lemma 5.6 to prove these.) Suppose now X? < A < Af°. Then by (5.13) we know that {u\ : e > 0} is bounded in L°°(f2). By elliptic regularity and the Sobolev imbedding theorem we see that {ucx : e > 0} is compact in C x (fi). Since e —* uex(x) is decreasing, (5.13) implies that ul(x) := lime^oWAl2-) e x i s t s for all x £ fi. The above compactness conclusion then implies that u\ —> u° in C 1 (fi) and furthermore, ul is a positive solution of (5.12) with e = 0. Therefore

76

Maximum

Principles

and

Applications

we must have vPx — u\, due to uniqueness. This proves conclusion (i) of the theorem. We next prove conclusion (ii). So we assume that A > Aj 0 . Let ma = min ux(x) = ux(xe),

xe G fio-

We claim that m e —> oo as e —> 0. Clearly this implies part (a). We prove this claim by an indirect argument and divide the proof into several steps. S t e p 1. / / me < M for some constant M and all e > 0, then d(x£,dflo) —* 0. Since A > X1 ° ((/>), we must have ||w£||L~(n) —> oo as e —> 0, for otherwise ue increases to a positive solution of (5.12) with e = 0, contradicting the conclusions in Theorems 5.3 and 5.7. Let us now pick up a sequence e n —> 0, and define un = Un/||w„||oo, where un = uxn. We easily see that - A u „ = \un - [b(x) + en}\\un\\P^1upl,

un\dQ = 0.

It follows that —Au„ < Xun in Q. Therefore / \Dun\2dx < A / u2ndx < X\Q\, and {un} is bounded in W0' (£}). This implies, by similar considerations to the arguments used in Step 1 of the proof for Theorem 5.7, that subject to a subsequence, iin converges weakly in W0' (Q) and strongly in Lq(fl) (for all q > 1) to some u G W,J,2(fi), and u ^ 0. An application of Lemma 5.6 shows that un is bounded from above by U_x on Q+ := Q. \ f2o- From this we easily see that u = 0 on £l+. Thus, as d£l0 is smooth, u|n 0 e WQ'2(C10). If ||wn||<x> = u(xn), xn e fi, then by Bony's maximum principle (see Theorem 2.10), there exists a sequence ik —> xn such that limfe-^ Aun(xk) < 0 and hence, from the equation for un, we obtain 0 < \un(xn)

- [b(xn) +

en]un(xn)p.

It follows that enllwnll^ 1 < A. Hence we may assume that enll^nll^ 1 —> £ for some £ > 0. Now we multiply the equation for un by an arbitrary ip £ Co°(fi 0 ) and integrate over Cl0, and pass to the limit n —> oo, to obtain that / Du-Dipdx= Jn0

I (Aw Jn0

tup)il)dx.

The Logistic

Equation

77

That is to say that U\Q0 is a weak solution to -Aw = (A - £up-l)u,

u|ano=0.

By the weak Harnack inequality, we deduce u > 0 in fioFrom the equation for un, we see that —Awn is uniformly bounded on ilo- By standard interior Lp theory for elliptic equations (see Theorem A.26), we find that un is bounded in W2'q(fl') for any q > 1 and any compact subdomain fl' of fio- By the Sobolev imbedding theorem, we know that subject to a subsequence, un —> u in C 1 (fi'). As u > 0 in fio, and ||wn||oo —* oo, we find that wn(a;) —> oo uniformly on any compact subset of fio- As e —+ w^ is monotone, u\ —> oo uniformly on any compact subset of fl0 as e —> 0. Thus we must have d(xe, dflo) —> 0 as e —* 0. S t e p 2. ijf m £ < M for some M and all e > 0, then {du\(xe)/di/e} is bounded from above, where ve is a unit vector in R to be specified later. It suffices to show that for any sequence e„ —• 0, {du€^(xen)/dven} has a subsequence which is bounded from above. Let us denote un — uef, xn = x€n

and Qn = {x e fio •' d(x,dflo)

>

d(xn,di}o)}.

Note that if xn € dQ.Q, then Q,n = QQ, and if fln is different from fio, then for large n, it is close to Q,Q by Step 1. Thus for any Q' CC fio, fi' CC f2„ for all large n. We may assume that 0 < e„ < 1 for all n. Then clearly un is an upper solution to the problem - A w = Aw- [b{x) + l\up in (l\fln,

U\9Q = 0,

w|an„ = un(xn),

(5.14)

and 0 is a lower solution. Therefore (5.14) has a positive solution vn satisfying 0 < v„ < un in fl \ fln. As un(xn) = vn(xn), it follows that dun(xn)/dvn

<

dvn(xn)/dvn,

where vn is the unit normal vector of dfln at xn pointing inward of fln. Thus it suffices to show that dvn{xn)/dvn is bounded. Clearly C0 := max{A 1 /( p ~ 1 ), M} is an upper solution to (5.14). By Lemma 5.6 we conclude that vn < CQ. This implies that — Avn has an L°° bound on 11 \ fln which is independent of n. Since furthermore, (1) vn\dnn is a constant which has a bound independent of n, and (2) for all large n, dfln is as smooth as fio with the smoothness not depending on n, by the Lp theory of elliptic equations up to the boundary (see Theorem A.27 and its proof in [Gilbarg-Trudinger]), we see that, for any q > 1,

78

Maximum

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and

Applications

\\vn\\w2'i(Q\Un) n a s a bound independent of n. By Sobolev imbedding theorems and the uniform smoothness of fi„, this implies that H^nllculnn,,) has a bound independent of n. In particular {\Dvn(xn)\} is bounded, and thus {dvn(xn)/dvn} is bounded, as required. Step 3. m 6 —> oo as e —• 0. Otherwise we can find a sequence e n —> 0 such that m £ii is bounded. By Step 2, {dun(xn) / dvn) is bounded from above, where vn is the unit normal vector of dfln at xn pointing inward of fi„. Here we follow the notations in Step 2. We show that this is impossible, and hence proving the claim. For all large n, dQ.n is as smooth as dtto and hence it satisfies a uniform interior ball condition: There exists R > 0 such that for any large n and x £ dQn, one can find a closed ball Bx of radius R such that Bx C fln and Bx n dQn = {x}. Let yn denote the center of BXn and define iP{x) = e-°lx-y^2

-

e~°R2,

where a is a positive number to be specified. We may assume that en < 1 for all n. Then, for any constant c satisfying 1 < c < e n and x £ BXn \ Bn, n where B = {x : \x — yn\ < R/2}, we have A[un(xn) + cip] + X[u„(x„) + cip] - en[un(xn) + cip]p > ce-^x-y^2[4a2\x - yn\2 - 2Na] - e „ c > „ ( x n ) / c + V] p R 2 > ce-° \a R? - 2Na) - [un(xn) + VF >0, if a, c and n are large enough. We fix a at such a value. Choose a compact set if CC fio such that K D U^=lBn. By the proof of Step 1, un —> oo on K. Hence we can find a sequence cn —> oo satisfying c-n < e n and un(x) > M + c n V|aS", for all x £ 9 5 " c X. Thus, un is an upper solution to the problem - A M — Xu-enup

in BXn\Bn,

u\dBxn

- un(xn),

u\dB* =

un(xn)+cntp\dB"•

By our choice of a, for all large n, un{xn) + cnip is a lower solution to this problem. Using Lemma 5.6 we deduce un > un(xn) + cnip in BXn \Bn, and it follows that dun(xn)/dun

> cndil>{xn)/dvn

= cn2oRe~°R

—» oo.

The Logistic

Equation

79

This contradicts the conclusion in Step 2. Thus the claim and hence part (a) in conclusion (ii) of the theorem is proved. It remains to prove part (b). Suppose that e n is a sequence of positive numbers decreasing to 0 as n —» oo, and un = uex . By the above proved part (a), we see that wn|9fj0 —* °° uniformly a s n - » oo. By Lemma 5.6 we deduce un < un+\ < U_\- Therefore un —> uo < U_x as n —+ oo. It follows that uo is a positive solution of (5.4). Since U_x is the minimal positive solution, we must have u 0 = U_x. The proof is complete. • In order to better understand the profile of uex, we consider wx := e ~ u\. It is easily seen that wx is the unique positive solution of the problem p 1

- A w = \w - [1 + e-1b{x)\w'p, w\dQ = 0.

(5.15)

If A € (Af, Af °), then by Theorem 5.9 (i), we see that wx —> 0 uniformly in Q, as e —» 0. We now consider the case that A > Xx °. If we denote by 6\ the unique positive solution of —Aw = Aw — w p , w|afj = 0, then by Lemma 5.6 we see that wex < 9\. Also by Lemma 5.6, we find that w\ is non-increasing with e. Therefore wQx{x) = lim £ _ 0 w\(x) € [0,9\(x)] exists. Furthermore, on any compact subset K of Clo, —Awex •=• Xwex — (wx)p has an L°° bound from above independent of e. By the Lp estimates and Sobolev imbedding theorem we find that w€x converges to w° in Cl(K). By Theorem 5.9 (ii)(b), we see that w\ —> 0 uniformly on any compact subset of H \ Q 0 . It follows that w°x = 0 over fi \ fio • Let #° denote the unique positive solution of -Aw = Xw-wp,

w\dno = 0.

(5.16)

By Lemma 5.6 we obtain w^ > 9\. Therefore w°x > 9\* in Cl0. We show that w° = 9^ in Q,Q. Indeed, from the inequality —Awx < Xwex < X9\ we deduce that {w\} is bounded in Wg1,2(fi) and therefore by a compactness argument, wex —> w° weakly in WQ' (fl) and strongly in Lq(fl) for any q > 1 (we also use the fact that ||iu^||oo is bounded). Since w° = 0 in fi+ and d£lo is smooth, we conclude that w°|n 0 G W0' (QQ). It follows easily that w°|o 0 is a week positive solution of (5.16). By standard elliptic regularity, w° is also a classical positive solution. But #° is the unique such solution. Therefore w°x = 9°x in fi0-

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We now find that u>° is a continuous function in 17, and as e —> 0, w\ —> u>° in Lq(Q), and the convergence wex(x) —> u>°(:r) is uniform on any compact subset of fi \ dfio- We claim that this convergence is uniform over CI. From the above discussions, it is clear that we need only prove the following conclusion: For any given S > 0, there exists (To > 0 so that w\(x) < <5Ve G (0,CTO), Vx e 5ff0 := {x G Q : d(x,dn0)

(5.17)

Denote $! A^° and tt0 C £la, we have A > A^'. Therefore the problem —Aw = \w — wp, w\ga„ = 0 has a unique positive solution 8X. If we extend 9ax to be 0 outside fla, then a simple compactness argument shows that 9X —> #° as a —> 0 uniformly in fi. In particular, we can find <j\ > 0 small so that A > Xx "x and 0 ^ (x) < 6/2 Vx G Sai.

(5.18)

On the other hand, let b(x) < b(x) be a continuous function such that b(x) = 0 on flai and 6(x) > 0 on fl \ Qai. Then by what has been proved above, we have wx —> 0J 1 uniformly on fiCTl/2, where w | is the unique positive solution to - A w = Aw - [1 + e~lb{x)\wp,

W\9Q = 0.

By Lemma 5.6 we deduce wx < w\. Choose <JQ < a\/2 such that for e < (To, w\<9^+6/2 Therefore, by (5.18), for e e

(0,
v>i<w\<

on

Vtai/2.

and x e S„0 C Sai/2, 0 ? + 6/2 < 6,

that is, (5.17) holds. We have thus proved the following theorem. Theorem 5.10 Suppose that A > A ^ . Then the unique positive solution w \ °f (5.15) converges uniformly to 9\ on Q. as e —* 0, where 0° is the unique positive solution of (5.16), extended to be 0 outside Qo. Let us observe that Theorem 5.10 gives a clear description of the profile of w\ for small e > 0: It is close to 0 over fl \ fio and close to a definite positive function 0° over Qg. It can be shown that when A = A^°, uex —* 0 uniformly in fi as e -» 0; we leave the proof of this fact to the reader.

The Logistic

Equation

81

Further related results to Theorems 5.9 and 5.10 can be found in [DuLi(2005)] and [Dong-Lu(2003)J.

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Chapter 6

Boundary Blow-Up Problems

In this chapter, we discuss boundary blow-up problems of the standard form Au = f(u)

in Q,

U\Q(I

= co,

(6.1)

and some of its variations. Here fi C RN is a bounded domain with Lipschitz continuous boundary, and f(s) is a continuous function of s € R1. By a solution to (6.1) we mean a function u € C ^ Q ) such that u(x) —> oo as d(x) = d(x, dQ) —> 0, and / Du • Dcpdx + [ f(u)dx = 0

V<£ £ C£°(n).

From standard regularity theory for elliptic operators, we know that such a solution belongs to C 2 (fi) if / is locally Lipschitz continuous. The study of (6.1) has a long history. Motivated by a geometric problem, L. Bieberbach considered, in [Bieberbach(1916)], the case N = 2, f(u) = eu, and proved that (6.1) has a unique solution u £ C2(Q), and u(x) - \nd(x)-2

= O(l) as d(x) -» 0.

H. Rademacher in [Rademacher(1943)] (motivated by a physical problem) proved that the same result holds when N = 3. Again motivated by geometric problems, C. Loewner and L. Nirenberg in [Loewner-Nirenberg(1974)] considered the case N >3 and f{u) = u^N+2~>^N~2\ and proved that (6.1) has a unique positive solution, and u(x)d(x)(N-W2

-> [N(N - 2 ) / 4 p - 2 > / 4 as d(x) -* 0.

In a more general setting, J.B. Keller [Keller(1957)] and R. Osserman [Osserman(1957)] independently proved the following result: 83

84

Maximum

Principles

and

Applications

Let N > 1, f(u) positive, non-decreasing, JQ F(t)~1/2dt < oo, where F(t) = / J /(s)rfs. Then (6.1) has at least one solution. Since the 1990's, the problems regarding existence, uniqueness, asymptotic behavior and multiplicity of solutions for (6.1) and its many variations have been extensively studied, and new applications of these problems have been found. We will discuss some of these results in this chapter. 6.1

The Keller-Osserman result and its generalizations

Consider the problem Au = /(«) in

fi,

(6.2)

where Q is a bounded or unbounded domain in RN, f(u) is a continuous function of u £ R1. Theorem 6.1 (The Keller-Osserman Theorem) Suppose that u Wj ' (CI) fl C(fi) satisfies (6.2) in the weak sense, namely, - [ Du- D<j>dx = J f{u)<j>dx, V<^ €

£

C^(fl),

and there exist a continuous nondecreasing function h(u) such that r°°

f(u) > h(u) > 0 in Rl, and I

Jo

r

r*

-1-1/2

I h(s)ds

'-Jo

dt < oo.

(6.3)

J

Then there exists a decreasing function g(R) determined by h such that u(x) < g(R) ifBR(x)

:= {y e RN : \y - x\ < R} C

fi.

(6.4)

Moreover, \img(t) = oo, lim g(t) = — oo. t—>0

(6.5)

t—>oo

Proof. Suppose that u is a solution of (6.2) and B — BR(X0) C fl. For each a > a® := max xe gB u(x), we consider the auxiliary problem Av = h(v) in B, v\dB = a.

(6.6)

Clearly u is a lower solution and any large constant C satisfying C > u in B and h(C) > 0 is an upper solution for (6.6). Therefore we can apply Theorem 4.9 to conclude that (6.6) has a solution v satisfying u < v < C in B. Since h(u) is non-decreasing, we can apply the maximum principle

Boundary

Blow-Up

Problems

85

as in Step 1 of the proof of Theorem 4.3 to conclude that (6.6) has at most one solution. Therefore v is the unique solution, and we denote it by va when its dependence on a is emphasized. This implies that v is radially symmetric for otherwise a different solution could be obtained by rotating v. Therefore v(x) = v(r),r = \x — XQ\, and v" + (N-

l)r~ V = h(v), v'(0) = 0, v(R) = a.

(6.7)

A simple upper and lower solution consideration, together with the uniqueness of va and the strong maximum principle, shows that vai > va2 in B whenever cx\ > a.2It is convenient to rewrite (6.7) in the form (rN-lv')'

=

^-^(v).

Integrating this identity from 0 to r yields v'{r)

= rl~N

f sN-1h[v(s)}ds Jo

> 0.

Therefore v(r) is an increasing function and v'{r) < r 1_JV /i[t;(r)] / / ^ r f s = Jo

{r/N)h[v{r)}.

It follows from (6.7) that v" = h[v(r)] -{N-

ly-^'ir)

>

(1/N)h[(v(r)}.

Therefore h(v) > v" > (1/N)h{v).

(6.8)

We now multiply (6.8) by v' and integrate from 0 to r to obtain H(v{r),v(°))

> v'(r)2

>

(l/N)H(v(r),v(0)),

where H(t, s) = 2 f h(^)d^. From this we deduce

Jv(0)

Jv(0)

Therefore r Jva(0)

H(£,va{0)rl'2d£

N1'2

[" Jva(0)

H(?,va{0))-l'2dH.

(6.9)

86

Maximum

Principles

and

Applications

We already know that va is increasing in a, therefore ^oo(O) := lima_,oo ^a(O) is either oo or a finite number. Were it oo we would have /•a

R < N1'2

/«oo

H{£, v a (0))- 1 / 2 de < N1'2

/

H(£,^(O))"1/2^,

/

Jva(0)

Jva(Q)

which is a contradiction because we can prove the following: /•OO

lim /

H($,ri)-1'idt

= 0.

To prove the above conclusion, we observe that, for any 77 > 0, due to the monotonicity of h, H{r] + t,rj)> H(t,0),

H(r) + t,rj)> 2h(r])t, Vt > 0.

Therefore, for any fixed M > 0, /•OO

/*00

Jn

JO

rM

= /

/»00

H{rl + d,r,)-1/2dZ+

Hfr +

JO r-M

CvrWdt

JM /*oo

< Jo / [2M»?)]-1/2r1/2de+/JM H(tor1/2dz /•OO

= [2/i(77)]- 1 / 2 2M 1 / 2 + /

fl-^.O)-1/2^.

Our assumptions on h(s) imply that h(s) —• 00 as s —* 00. Therefore /•oc

/»oo

l i m ^ / H(tvr1/2dt< Jrt

H{S,0)-V2d£. JM

Letting M —> 00, we obtain what we claimed due to (6.3). Therefore t>co(0) is finite. It is evident that i>oo(0) is determined by both h and R. Denote g(R) = i>oo(0). From u < va in B we deduce U(:EO) < fl(-R). We next show that g(R) is decreasing in R and has the properties in (6.5). To show that g{R) is decreasing, we denote the unique solution of (6.7) by vaiR. We already know that v'aR(r) > 0 for r <E [0,R]. Suppose i?i < i?2! then a = vatR2(R2)

>

va,R2(Ri).

Boundary

Blow-Up

Problems

87

Therefore va,R2 is a lower solution for (6.7) with R = R\. As large constants are upper solutions, the unique solution vcttR1 must satisfy vcttRl > va 5(^2)- We will later show further that g(Ri) > 5(^2)Let 5(0) := limt-^o d{t)- By the monotonicity of g{t) we know this limit is either finite or 00. If g(0) is finite, then from (6.9) we obtain /•OO

H(t,g(0))-1'2d$

/

= 0.

Jg(0)

This is impossible since H(£,g(0)) is positive for £ > 5(0). Therefore g(0) = 00. Similarly 5(00) := limt_>oo9(£) is either finite or —00 due to the monotonicity of g(t). Were it finite, we would deduce from (6.9) that /•oc

co=/

H^gioo))-1'2^.

Jg(oo)

This is impossible since by (6.3), in case g(oo) < 0, we have ^,5(oo))-1/2^ Jq /•0

<

//(^(oo))"1/2^ /

Jfl(oo)

poo

F^O)"1/2^-^;

J0

and in case 5(00) > 0, we have /•OO

/*00

Jg(oo)

JQ

It remain to show that g{R\) > 9(^2) when R\ < i?2- We first show that for any fixed R > 0, v^^r) := lima^oo Daifl(r) is well-defined. Indeed, from what is proved above, we find that for each r G [0, R), va
~ v*A0)

= / ^ Jo

/ Jo

^-^[v^R^dtds,

that ^00,R satisfies the same integral identity and hence VOO.R solves (6.6) with a = 00. From our earlier argument, we know that v<XtR1 (r) > vOCtR2 (r) for r 6 [0, Ri). Were g(iii) = 5(^2), we would have, for w = V00IR1 —V00^R2, Aw = /i^oc-RiOO] - Mwoo,.R2(r)] > 0, w > 0 in 5 f l l , and w(0) = 0. This implies, by the strong maximum principle, that w = 0, which is a contradiction. •

88

Maximum

Principles

and

Applications

In the above proof of the existence of a unique solution to (6.6), the assumption that (6.2) has a weak solution u 6 Wio'c (fi) n C(fi) was used. Such an assumption is unnecessary for this purpose. In fact, (6.6) always has a unique solution when a > 0. To see this, we let ip be the unique solution of AV> = 1 in B, ip\9B = 0. Then ip < 0 in B and for C, > h(0), we have A(C^) = C > MO) > MCVO in B, 0/»|ae = 0. Therefore £ip is a lower solution of (6.6) with a > 0. As before, any constant C > a is an upper solution, and C > (ip holds in B. Therefore (6.6) has a solution, which is unique due to the mono tonicity of h. Now from the last part of the proof of Theorem 6.1, we find that under the conditions for h in Theorem 6.1, the problem Av = h(v) in BR(0),

v\dBR{0)

= oo

(6.10)

always has a solution v(x) = v(r), r = \x\. Let us observe that in the case N = 1, (6.10) has a unique solution. Indeed, in this case (6.8) red-uces to v" = h{v) and hence rv{r)

r = /

H(S,v(0))-1/2<%-

(6.11)

Jv(0)

Letting r —> R we obtain /•oo

R=

/

H(t,v(0))-1/2<%,

Jv(0)

or equivalently /•OO

R = Jo

2 / /i[u(0) + s]rfs J L Jo

-1

Since h(s) is nondecreasing and not a constant function when s is large, the above identity uniquely determines v(0), and then we can use (6.11) to uniquely determine v(r) for any r G (0, R). This shows that the solution of (6.10) is unique. Whether this uniqueness result remains valid for iV > 2 is an open question. However, we have the following result.

Boundary

Blow-Up

89

Problems

P r o p o s i t i o n 6.2 Under the assumptions for h in Theorem 6.1, suppose further that h is locally Lipschitz continuous. Then for any N > 2 and almost every R > 0, problem (6.10) has a unique solution. Moreover, if (6.10) has more than one solution for some RQ > 0, then it has a continuum of solutions on BR0 (0). Proof. We will need some arguments in the proof of Theorem 6.1 and in the above discussion after that proof. We denote the unique solution of (6.6) with B = BR(0) and a > 0 by v R,a- We already know that UR := lim vRta is a solution to (6.10). We show that UR is the minimal solution to (6.10). Let u be an arbitrary solution to (6.10). Then for each a > 0 we can find Ra e (0,R) such that VR,a(x) < u(x) when |x| G (Ra,R) due to the behavior of u near 9 5 ^ ( 0 ) . Since h(s) is non-decreasing with s, this implies, by the maximum principle, v R,ai.x) S u{x) in BR{fS). It follows that uR(x) = lima—joo VR,a{x) < U(x) in BR(0). Therefore UR is the minimal solution to (6.10). Since each VRtCt is radially symmetric, so is UR. We next prove that (6.10) has a maximal solution. We first observe that UR(X) is non-increasing with R; that is, R\ < R2 implies UR1(X) > UR2(X) for all x £ BR1 (0). Indeed, let u be any solution of (6.10) with R = R2, then we can find Ro G (Q,Ri) such that u(x) < UR1(X) when Ro < \x\ < Ri. Thus, as before the maximum principle implies u(x) < UR1(X) on BR1(0). In particular UR2(X) < UR1(X) in BR1(0), as claimed. Moreover, UR{x)= lim uR,{x), i 6 B f i ( 0 ) (6.12) R'—*R—Q

is well-defined and UR(X) > u(x) in BR(0) for any solution u of (6.10). A standard regularity consideration shows that UR is a solution to (6.10), hence the maximal solution. UR(X) is radially symmetric because it is the limit of such functions. UR(X) is non-increasing in R, since R\ < R2 implies URl(x) =

lim ix —> tl\—U

uR,(x) >

lim

UR,+{R2_RI)(X)

= UR2(X),

VZ e

BRl{0).

i t —>t\.\—U

A similar consideration shows that lim#<_,#+() UR>(X) is a minimal solution to (6.10) and thus necessarily UR{X) =

lim

UR>(X).

Maximum

90

Principles and

Applications

Analogously, \imR>->R+oUR>(x) is a minimal solution of (6.10) and limR/->R-o URI(X) is a maximal solution of (6.10). Hence

Since h(s) is locally Lipschitz continuous, by Proposition A4 in [FranchiLanconelli-Serrin(1996)], for any £ G (—00,00), the initial value problem u" +

N

-1

u' - h(u) = 0, «(0) = £, u'(0) = 0

(6.14)

has a unique solution as long as the solution exists. Thus, whenever UR(0) = UR(0), we must have Wfi(r) = UR{T) which implies that (6.10) has a unique solution. By (6.12) and (6.13), we find that URO(0) = URO(0) whenever UR(0) is continuous at RQ. From the proof of Theorem 6.1, we see that R —> UR(0) is a decreasing function, and hence it is continuous almost everywhere. Thus (6.10) has a unique solution for almost every R > 0. If (6.10) has more than one solution for some R = R0, then by what has just been proved, we must have URO(0) > URO(0). Consider now the initial value problem (6.14) with £ G (URO(0),URO(0)). It has a unique solution v^(r) defined in some small interval [0,ro). Since h(s) is non-decreasing, vg(r) must stay between UR0(r) and UR0(r) and therefore it is denned in [0, R0) and thus solves (6.10). Hence (6.10) has a continuum of radially symmetric solutions. The proof is complete. • Corollary 6.3 Suppose that h(u) is a positive, non-decreasing function in Rl and satisfies the integral inequality in (6.3). Then for any bounded domain Q, C RN with Lipschitz continuous boundary dfl, the problem Au = h(u) in $7, u\ga — 00,

(6.15)

has a minimal solution u t and a maximal solution u* in the sense that any solution of (6.15) satisfies M„ 0 we consider the problem Av = h(v) in $7, V\QQ = a.

(6.16)

As before, if tp denotes the unique solution of A ^ = 1 in n, ip\da = 0, and if C > h(Q), then (ijj < 0 in Q and is a lower solution of (6.16). Any constant C > a is an upper solution of (6.16) and satisfies C > Qip in fl.

Boundary Blow-Up Problems

91

Therefore (6.16) has a solution va, which is unique due to the monotonicity of h. Moreover, we can use the monotonicity of h and the uniqueness of va to easily deduce that va is increasing in a. For any x € fi, by Theorem 6.1, we have va(x) < g[d(x,dCl)/2] and hence u*(x) := limQ_>oo va(x) is finite. Moreover, for any compact subset K of £1, (ip < va < g(6x) in a small neighborhood NK of K, where 8K = (l/2)dist(K,dQ). Therefore p Ava is uniformly bounded in iV^-, and by the L theory va has a bound in W2'P(K) (Vp > 1) for any compact subset K of 0.. It follows that va —> u* in Cl(K) and hence u„ satisfies in the weak sense Au* — h(u*) in CI. Since va increases to M* as a —> oo, we have «, > va in fi for any a > 0. Therefore U*|df2 = OO.

We show that u» is a minimal solution of (6.15). Indeed, let u be any solution to (6.15). Since u(x) —> oo as d(x,dfl) —> 0, for any given a > 0, we can find 5 = Sa > 0 small enough such that u(a;) > va(x) for cc G fls '•= {x € fi : d(x,dQ.) < 5}. Then we use the monotonicity of h and apply the maximum principle as before and deduce that va 0, define fi5 = fi\$l,s. By what has been proved above we find that (6.15) with fi replaced by Q,s has a minimal solution wf. Moreover, by the same comparison argument used before, we easily see that u*1 > usr2 in nSl when 5\ > 62, and for any small S > 0 and any solution u of (6.15), we have M < u^ in fi"5. Therefore, u*{x) := lim,j^o wf (a;) exists for any i £ f l and u* > u in fi for any solution u of (6.15). Moreover, a standard regularity consideration as before shows that u* is a solution of (6.15). Therefore it is the maximal solution. D Remark 6.4 The monotonicity condition for h(u) in Corollary 6.3 cannot be dropped. By Theorem 4 of [Aftalion-Reichel(1997)], for any C2 convex domain fl, there exists a continuous function h(u) which is positive and satisfies the integral inequality in (6.3) but (6.15) has no solution. Remark 6.5 If h(u) is positive and nondecreasing in Rl and (6.15) has a solution «o in some bounded domain Q C RN, then we necessarily have r°°

r

/ Jo

r*

-1-1/2

/ h(s)ds

dt < 00.

'-Jo

Suppose by way of contradiction that f°°

/ Jo

r

/*'

T-i/2

/ h(s)ds L Jo -•

dt = 00.

(6.17)

Maximum

92

Principles

and

Applications

By Proposition Al in [Franchi-Lanconelli-Serrin(1996)], for any £ G R1, the initial value problem u" + {N-

l ) r - V = h(u), «(0) = f, u'(0) = 0

has a solution u(r) which exists as long as the solution remains bounded. We now fix £ > 0 such that £ > rninn UQ. The analysis leading to (6.8) gives u'(r) > 0 and ru(r)

/

j-u{r)

H{s,i)-1/2ds

N1'2

H(s,£)~1/2dS.

I

Since I

H(3,^)-^2d3>

H(3,0)-^2d3,

we see by (6.17) that u(r) remains bounded as long as r is bounded. Therefore u{r) is defined for all r > 0, uir) - t o o a s r - t o o and u(r) > u(0) = £ for r > 0. We define VQ(X) = u(\x — xo\), where XQ G D. satisfies uo(xo) — minnuo. Then Qo := {x £ Q : UQ(X) < v0(x)} is nonempty with its closure contained in fi. Moreover, A(w0 - v0) = h(u0) - h(v0) < 0 in fi0, (u0 - v0)\asi0 — °By the maximum principle we deduce «o > ^o in fio, which is a contradiction. It turns out that the condition h(u) > 0 in R1 is not necessary, as can be seen from the following result, which extends Corollary 6.3 to more general situations. Theorem 6.6 Suppose that f(u) is a continuous function in R1 and there exists so > 0 and a continuous nondecreasing function h(u) defined for u > so such that f°°

f(u) > h(u) >0foru>so,

r /"'

/ h(s)ds J So

1-1/2

dt < oo.

(6.18)

J So

If Q, is a bounded Lipschitz domain and there exists some v* G Wl'2{Q) n L°°(n) such that Av» > / ( f * ) in Q,

(6.19)

Boundary

Blow-Up

93

Problems

then (6.1) has at least one solution u 6 C 1 (f2) satisfying u > v* in Q. Moreover, it has a minimal solution «* and a maximal solution u* among all such solutions. Let us note t h a t if f(si) (6.19). Proof.

< 0 for some s i < so, then u* = s\ satisfies

For a n y a > ||w*||oo>

w e

consider t h e problem

Av = f(v) in CI, v\9ri = a.

(6.20)

By assumption, v* is a lower solution t o (6.20). Define Ca : = m a x { a , so}\ then Ca > v* a n d is a n upper solution of (6.20). Therefore we can apply Theorem 4.11 t o conclude t h a t (6.20) h a s a minimal weak solution va a n d a maximal weak solution va in t h e order interval [v*,Ca] c W1'2(fl). B y s t a n d a r d regularity result we know t h a t va, va € W2'p(Cl) for all p > 1 and hence they a r e C 1 in CI. We claim t h a t va a n d va a r e nondecreasing in a. Indeed, if ||u*||oo < ai < a2, then va2 is a n upper solution for t h e equation satisfied by vai a n d hence, by Lemma 4.10, v := m i n { v Q l , v Q 2 } is a weak upper solution of (6.20) with a = a\. Clearly v„ < v is again a lower solution. Therefore by Theorem 4.11, equation (6.20) with a = ai h a s a minimal solution vai in t h e order interval [v*,u]. In particular, vai G [v*,vai]. B u t by definition, vai is t h e minimal solution in this order interval. Hence vai = vai, which implies t h a t v = vai a n d va2 > vai. B y a similar consideration, we have vai > vai. For any XQ G fl, we now prove t h a t w*(a;o) = limQ_»oo va(xo) We choose R > 0 small a n d consider the problem Aw = h(w)

in BR{X0),

w\dBR{x0)

is finite.

( 6 -21)

= oo,

where h{u) = h(u + so) for u > 0 a n d h(u) = h(so) for u < 0. Clearly h satisfies all t h e conditions in Corollary 6.3 a n d hence (6.21) h a s a minimal solution % . Moreover, WR can be obtained as t h e limit of t h e solutions of (6.6) with h replaced by h a n d hence by the proof of Theorem 6.1, we find UJR(XQ) = m i n ^ R ^ ) WR —> oo as R —> 0. Therefore if R > 0 h a s been chosen small enough, then WR > 0 in BR(XO) a n d hence v* : = WR + so satisfies Av* = h{v*),

v* > s0 in BR(x0),

V*\QBR{XO)

= oo.

94

Maximum

Principles

and

Applications

We claim that va < v* in BR(X0). Otherwise, D := {x S BR(X0) : va(x) > v*(x)} is nonempty with its closure contained in BR(XQ). Therefore A(va - v*) = f(va) - h(v*) > h{va) - /»(«*) > 0 in D, (va - v*)\dD = 0. By the maximum principle, this implies that va — v* < 0 in D, which is a contradiction. Therefore va < v* in BR(x0) and hence u^{x0) is welldefined. The above arguments actually show that va is uniformly bounded in any compact subset of fi. Therefore, as before, we can use standard regularity theory to conclude that u , 6 C^fi) is a solution of (6.1). Similarly, we can show that u* := lim va

(6.22)

a—>oo

is well-defined, belongs to Cl(Q) and is a solution of (6.1). We next show that u* is the minimal solution of (6.1) among all its solutions satisfying u > v*. Let u £ C 1 (fi) be any solution of (6.1) satisfying u > v*. Denote va = min{w,v a }, where va is the solution of (6.20) with a > ||v*||oo obtained above. For any small 5 > 0, we define fls = {x G CI : d(x, dfl) > 6} and consider the problem AM = f(u) in Cls, u\9Q6 = va.

(6.23)

Then v * ^ < va\^s form a pair of a weak lower and upper solutions for (6.23). Therefore, by Theorem 4.11, there is a minimal solution u5 satisfying v* < u5 < va in £ls. By adapting the argument used to show the monotonicity of va in a, we see that us is non-increasing in 5 and therefore u := l i m ^ o us is well-defined in 0 and satisfies v» < u < va < va. Moreover, by the regularity theory for elliptic operators, it is easily seen that u is a weak solution to (6.20). Since va is the minimal solution in [v*,v a ], we necessarily have u = va. This implies that iia — va, and hence u > va. Therefore u > u*. This proves that w* is the minimal solution. For Jl* defined above, we now consider the problem AM = f(u) in Qs, u\dQS = oo.

(6.24)

Applying what has just been proved for (6.1) to (6.24), we find that (6.24) has a solution u* satisfying u* > v* in tts, and M* is given by the limit in (6.22) with va replaced by v°, where vf is the maximal solution of (6.20) over Q,s in the order interval [u*|n,s,CQ]. Suppose that u £ Cl{fi) is any solution of (6.1) satisfying u > v*. Since u is uniformly bounded in Qs, for all large a, u < a in fls and hence U\QS is a lower solution of (6.20) over Cls.

Boundary Blow-Up

Problems

95

It follows that max{w|fj,5, v%} < Ca form a pair of weak lower and upper solutions of (6.20) over Qs. By Theorem 4.9, we conclude that it has a solution u satisfying ma,x{u\ns,v£} < u < Ca. In particular, v" < u < Ca. But vf is the maximal solution in this order interval. Hence we must have u = v£, which implies u < vf in Qs. Therefore, u < u5f. Repeating our argument used for us, we find that v" is non-increasing in S. It follows that ui is non-increasing in 5 and u* := lim,5_,o uf is well-defined. A standard regularity consideration shows that u* is a solution of (6.1). From u < u* we obtain u < u*. Therefore u* is the maximal solution. • Remark 6.7 The results in this section can be extended to the case where A is replaced by the so called p-Laplacian operator A p ; this has been done except for Theorem 6.6, see [Du-Guo(2002)] and the references therein for more details.

6.2

Blow-up rate and uniqueness

We consider in this section the asymptotic behavior near the boundary for boundary blow-up solutions of (6.1) and the question of uniqueness of such solutions. Theorem 6.8 Suppose that the conditions in Theorem 6.6 are satisfied so that (6.1) has a solution u. Moreover, suppose that dVt is C2. Then lim « d{x)-^a d(x)

= !,

(6.25)

where -1/2

d{x) = d(x,dn),

*(u) = 4 = /

[/

f(s)ds

dt.

Proof. Let BR be a ball of radius R contained in fi. Then (6.19) over BR is satisfied by V*\BR. Therefore we can apply Theorem 6.6 to conclude that Av = f(v) in

BR,

V\9BR

= oo

(6.26)

has a maximal solution v. v must be radially symmetric about the center of BR for otherwise a rotation of v gives a solution of (6.26) which is not comparable with v. Therefore we may write v = v(r), r e [0, R).

Maximum

96

Principles

and

Applications

We claim that v > so in BR if R is small enough. To see this, we consider the initial value problem u" + {N-

l ) r - V = / ( « ) , u(0) = s 0 , u'(0) = 0.

As in Remark 6.5, by Proposition Al in [Franchi-Lanconelli-Serrin(1996)] we know that a solution VQ(T) to this problem exists as long as the solution remains bounded. As in Remark 6.5 we deduce v'0(r) > 0 and rv (r) 0{r) rvQ

/

_

_

F{s, s0)-^2ds

Nl/2

rv0(r)

_

J Sn

/

F{s,

s0)-1/2ds,

JSo 'so

where F(t,s) = Js f{u)du. By (6.18), we deduce fV0(r) pVQ(r)

/

roc roc

l/2

F{s,s0)- ds<

J sn

/ J sn

r-s />s

"1-1/2

[ / h(t)dt

ds < oo.

J so

This implies that there exists RQ < oo such that vo(r) is defined in [0,R0), and it increases to oo as r —> RQ. Now for R € (0, RQ) we can apply Theorem 6.6 over BR with v* replaced by max{v*,fo} to see that the maximal solution v satisfies v > vo > so in BR. (In fact, we can adapt the argument in the proof of Theorem 6.1 to show that v(0) —> oo as R —> 0.) We next consider the problem Aw = f(w) in

BR\B~RI,

W\9BR

= si, w\aBttl

= oo,

(6.27)

where R\ G (0,R), «i > v0(R). This problem has a solution w satisfying w > VQ, which can be obtained as the limit of wa as a —> oo, where wa > vo is the minimal solution of (6.27) with w\dBR = °° replaced by v\dBRl

= Oi >

V0(Rl).

In order to understand the asymptotic behavior of an arbitrary solution u of (6.1), we first study the behavior of v and w constructed above. We will show that Mr^RMv(r)]/{R-r)>l, Mr^Rl^[w{r)]/(r

- Ri) < 1.

Since v(r) satisfies

(r^W)'

= r^-VMr)), „'(0) - 0,

(6.28) (6.29)

Boundary Blow-Up Problems

97

we have rN-

sN-1f(v(s))ds>0{oTre(0,R),

"V(r) = f Jo

and [(rw-V)2]' =

2r2^N-1\'{r)f{v(r)).

It follows that, for r0 < r < R, r^-^v'ir)2

=r20(N-1}v'(r0)2

s2(N^

+2 f

f(v(s))v'(s)ds

Jro

= r^-Mro)2

2r^N-^F(v(r),v(ro))

+

[s2(N-V-r2(N-V]f(v(s))v'(s)ds.

+2 f Jro

Therefore, for TQ < r < R, v'(r)2 = G0(r)+2F0(v(r))

- 2F0(v(r0)) + 2 6 ^ ) ,

where G0(r) = ( r o / r ) 2 * " - 1 ) * ' ^ ) 2 , F0(t) = F(t,s0) [(s/r)2^N^

Gi(r) = f

-

and

l]f(v(s))v'(s)ds.

Clearly, for ro < r < R, 0 > G^r) = f

[{s/r)2^-1^

-

l}f(v{s))v'(s)ds

Jro

> - [ 1 - ( r o / r ) 2 ^ - 1 ' ] f /(u( S ))«'( s )d S Jr0 >-[l(r0/R)2{N-1}][F0(v(r)) - F0(v(r0))}. The conditions in (6.18) imply that f(u) i7o(v(^)) - • oo as r -> i?. It follows that

x,'( r ) 2

„'( r )2 1

hm R

* - 2W^))

—> oo as u —> oo. /roX2(iv-i)

" ^ * 2J%£M) " ^

Since r 0 G (0, R) is arbitrary, we deduce i/(r) 2 r^fi 2F 0 (t;(r))

'

Therefore

•

98

Maximum

Principles

and

Applications

Therefore, for any given small e > 0 we can find Ro G (0, R) such that v'(r)2 > (1 - e)22F0(v(r)),

Vr G

(R0,R).

From this we obtain rOO

{2Fo(s)}-V2ds Is

*(v(r)) = / Jv(r)

[2F0(v(S))}-^2v'(s)ds

= J

>(l-e)(R-r),

Vre{R0,R).

Therefore ljmr^R*(v(r))/(R

- r ) > 1 - e.

Since e > 0 is arbitrary, this implies (6.28). To show (6.29), we observe from w(r) —> oo as r —* Ri that there exists ri G (Ri,R) such that w'(ri) < 0 and hence, by integrating from r G (R%,ri) to ri of the identity

(^-V/^-'/fa,), we obtain - r1-"

w'{r) = {n/r^^w'in)

f ' aN-1f(w{a))ds

< 0, Vr 6 (iZi.n).

Now for i?i < r < ro < ri we have, by repeating the above analysis for v, ro

/ = rl{N-l)W'{rQ)2

+

s2(N-Vf(w(s))w'(s)ds

2r2^N-^F(w(r),w(r0))

ro

/

[3*W-»

-r2lN-V]f(w(s))w'(s)ds,

from which we deduce w'(r)2 hm —=—,——— = 1. r-*Rx 2F0(w(r)) Therefore, for any given small e > 0, we can find R0 G (i?i, r 0 ) such that -w'(r)

< {l + €)[2F0(w(r))]^2,

Vr G

(RuRo).

Boundary

Blow-Up

Problems

99

From this we obtain /•OO

*(w(r)) = /

[2Fo(s))-1/2ds

Jw(r)

= f

1

[2F0(w(s))}-1/2w'(s)ds

< (1 + e)(r - i?i),

Vre(RuRo).

Therefore I l S r - ^ t f (u>(r))/(r - i2i) < 1 + e. Since e > 0 is arbitrary, this implies (6.29). We are now ready to prove (6.25). Since u(x) —> oo as d{x) —> 0, we can find 6 > 0 such that u(:r) > so when d(x) < S. As dfl is C 2 , it satisfies a uniform interior and exterior sphere condition, namely, there exists some R„ > 0 small so that for any x 6 dfl there exists a ball B1 of radius 72* such that Bl is contained in f2 and touches <9f2 at x; there also exists a ball B 2 of radius i?» so that B 2 is outside fl and touches 0, ve(x) = v(\x — ye\) and w*(x) = w(\x— y*\), where ye is a point on the line passing through xf and yo such that yo is between ye and a;, and \yo — yc\ = e. We want to show that v° > u in B, w* x* along the line segment /a,, with end points yo and a;*, we have l i m ^ ^ M a ^ M x ) ^Um^^.^di-yolMiZ-lar-i/ol) > 1 by (6.28), and limx-+Xt,$(u(x))/d(x)

< lhnx-,x,$!(w(\x

- y*\))/(\x - y,| - i?i) < 1

Maximum

100

Principles

and

Applications

by (6.29). Therefore lim

«

+x,,x€lx,

^

= 1-

d(X)

The above limit is uniform in x* € dCl, which implies (6.25). It remains to prove (6.30). Firstly we show that ve > u in Bc := {x £ RN :\x — yt\ < K}. Indeed, we can apply Theorem 6.6 over Be using u\et as v* in (6.19) to obtain a maximal blow-up solution ve over B€ satisfying ve > u in Be. It is easy to see that ve = ve. Therefore ve > u in Be. Letting e —> 0, we obtain v° > u in B. For each wa used above to define w, we define tu*(i) = wa{\x — y*\) and find u > w* in 9 B 2 n fi since si < u(x) when d(:r, <9fi) < 5. Moreover, u(x) > a tor x £ fin A close to dfl. Therefore wa '•= min{u(:r), w*(x)} = w^(x) in a small neighborhood of <9(f2 O A) in f2. If we extend wa to A by w* outside fi n A, we find that wa is a weak upper solution of the equation satisfied by wa, which has VQ(\X — y*|) as a lower solution and v0(\x — 2/*|) < si < w a in A Therefore we can find a solution w satisfying vo(\x — y*\) < w(x) < wa{x) for x e A Since w^ is the minimal such solution, we necessarily have w = w^ and hence w£ < u in fi n A. From this we obtain w* < u in A (Ifl. • The result in Theorem 6.8 can be easily extended to the case that A is replaced by the p-Laplacian A p ; such a result was proved in [Ma] under the extra restriction that / is nondecreasing. We next consider the uniqueness problem. This in general depends on a better understanding of the blowup behavior than (6.25), and can be achieved by further restricting the nonlinear function f(u). To emphasize the dependence of $ on / , we will use the notation ^ = tyf. From (6.25), we immediately obtain the following result. Corollary 6.9 furthermore,

Suppose that the conditions in Theorem 6.8 hold, and

lim^

0 0

- ^ > l V/?e(0,l).

(6.31)

Then any solution u of (6.1) satisfies lim ;,/ = 1, d(x)->o $f[d(x)] where $ / denotes the inverse oftyj.

(6.32) '

K

Boundary Blow-Up

Problems

101

Let us note that if there exist constants p > 1 and c > 0 such that Iim ^ t-»oo

= c > 0, tP

then one easily checks that (6.31) is satisfied. It is also satisfied if /(£) = e*. Theorem 6.10 Suppose that CI and f(u) satisfy the conditions in Theorem 6.8 and furthermore, (6.31) holds, /(0) = 0 and f(t)/t is nondecreasing for t > 0. Then (6.1) has a unique nonnegative solution. Proof. Since /(0) = 0, we may take u* = 0 in (6.19) and hence (6.1) has a minimal nonnegative solution u* and a maximal nonnegative solution u*. Moreover, (6.32) holds. It suffices to show that u* = u*. For fixed £ > 1, we have A(£«*) = £/(«*) < /(£«„) in SI. Due to (6.32) we find that £u*(x) > u*(x) when x is close to dQ. Therefore we can apply Lemma 5.6 to conclude that u* < £u* in fl. (Note that f(t)/t nondecreasing is enough in our situation here; this can be easily seen from the proof of Lemma 5.6.) Since £ > 1 is arbitrary, we deduce u* < u*. Therefore u* = u* and (6.1) has a unique nonnegative solution. • Remark 6.11 (i) Theorem 6.10 does not cover the case f(u) — eu. For this nonlinearity, a better estimate for the boundary blow-up rate can be obtained: u{x) — \zi[d{x)~2] is bounded. From this, it is easy to show that uniqueness holds, (ii) In [Lazer-McKenna(1994)], for a wide class of nonlinear functions f(t), including e*, f (p > 3), it is shown that (6.1) has a unique solution; moreover, if Z(t) satisfies Z"(t) = f(Z(i)) for all small t > 0 and Z(t) —> oo as t —> 0, then u(x) — Z(d(x)) —> 0 as d(x) —• 0. This estimate is not true for f(t) = t3, which becomes clear from higher order estimates; see [del Pino-Letelier(2002)], [Bandle(2003)] and [Bandle-Marcus(2004)] for some general higher order approximations of the blow-up rate. (iii) In many cases, uniqueness of boundary blow-up solutions can be obtained without knowing the exact blow-up rate; see [MarcusVeron(2003b)] and the next section for some examples. (iv) Problem (6.1) with A replaced by a general linear second order elliptic operator L was studied in [Bandle-Marcus(1995)].

102

Maximum

Principles

and

Applications

(v) If u is a solution of (6.1) with f{u) = eu, then v{x) = e"' 1 ' satisfies \Dv\2

Av =

J

!

2 •

o

i

—h v in S2, «asi — oo.

v This and more general boundary blow-up problems involving the gradient term Dv were studied by many people, see [LasryLions(1989)], [Bandle-Giarrusso(1996)], [Porreta(2004)] and the references therein for more details. 6.3

Logistic type equations with weights

In this section we consider the boundary blow-up problem (5.4) and related problems. In order to include more general cases, we rewrite this problem in the following form, - A u = a(x)u — b(x)up in fi, ulr^ = oo, u|r 0 = 0,

(6.33)

where f2 is a bounded domain in RN, Too is a nonempty open and closed subset of dfl, To = dfl \ Too (which might be empty), p > 1 is a constant, a is a continuous function on fl, b(x) is continuous and positive on fi. Let us note that b(x) is now allowed to vanish on part or all of dfl as well as to blow-up on part or all of d£l. By a solution to (6.33) we mean a function u G Cl{Q) such that [ Du-Dcf>dx= f [a(x)u - b(x)up]4>dx V ^ C 0 W ( ( ] ) Jo, Jn

(6.34)

and u{x) —> 0 as d(x,T0) —• 0, u(x) —> oo as d(x,Too) —> 0. For simplicity, in the following, we will denote d(x) = d(x, Too). Lemma 6.12 Suppose that b(x) > cd(x)a in D, for some constants c > 0 and a 6 R1. Then any solution of -Au = a(x)u - b(x)up, u > 0 m

fi,

(6.35)

satisfies supd(x)Pu(x) xen

< oo, where f3 = (2 + a)/(p — 1).

Proof. For any XQ € D, we denote d0 := d(x0). In the ball B0 := 5d 0 / 2 (x 0 ), b(x) > cd{x)a > CQCZQ, where CQ > 0 is independent of XQ.

Boundary

Blow-Up

Problems

103

Therefore —Aw < HallooU - cod^vP in BoLet u0(x) = {c0/^)ll{p'l)d^u{x0

+ (do/2)x) for x G B := B x (0); we obtain

- A M 0 < (^o/2)2||a||ooWo - UQ < Au0 - ug in B, where ^4 = maxfj(d(:r)/2)2||<2||oo- By Theorem 6.10, the problem -AU

= AU-Up

in B, U\dB = oo

has a unique positive solution U. Applying Lemma 5.6 we deduce UQ cd(x)a with a < —2, then P = (2 + u)/{p — 1) < 0 and hence, as a consequence of Lemma 6.12, any solution of (6.35) stays bounded near Too. Therefore, (6.33) cannot have a positive solution for such b{x). Theorem 6.14 Suppose that 80. is C2 and b(x) < cod(x)a for some Co > 0 and a > —2. Then (6.33) has a minimal positive solution u and a maximal positive solution u in the sense that any positive solution of (6.33) satisfies u 0 we consider the problem —Au = a{x)u — b(x)up in fl, u\roo = c, u|r 0 = 0.

(6.36)

We claim that (6.36) has a unique positive solution. We will use a lower and upper solution argument to prove this claim. To this end, we introduce, for some small but fixed 5 > 0, Qs := {x € RN : x € fi or d(x) < 6}, and extend a(x) to a continuous function over Qs. Let a* = l+maxfAj 2 , ||o||£~(n<)}. Then choose a nonnegative continuous function b*(x) on W such that b*(x) < b{x) on Q and ^ s {x £ W : b*{x) = 0} has C2 boundary and small volume so that Ax ° > a*. Then, by Lemma 5.6 and Theorem 5.7, there exists a unique positive solution u* for the problem —Au = a*u — b*{x)up in fls,

U\QQS

= 0.

Choose a large constant M > 1 such that Mu* > c on Too- Then it is easily checked that Mu* is an upper solution to (6.36). Clearly u = 0 is a

Maximum

104

Principles

and

Applications

lower solution to (6.36). Therefore, (6.36) has at least one positive solution. By Lemma 5.6, there is at most one positive solution. Hence (6.36) has a unique positive solution, which we denote by uc. Applying Lemma 5.6 again, we deduce that uc is nondecreasing in c. For any XQ € fi, we can find a small ball Br(x0) C fl such that b(x) > a > 0 over Br(xo). By Theorem 6.6, the problem - A u — Wa^u - auv in Br(x0),

u\dBr(Xo) = oo

has a positive solution u„. Applying Lemma 5.6 we find that uc < u* in BT{XQ). Therefore u(x) := linic-^oo uc(x) is finite for every x 6 fl. Moreover, a regularity consideration shows that uc converges to u in the C norm over any compact subset of 0, \ Too and u is a solution of (6.33). We claim that u is the minimal positive solution. Indeed, if u is any solution of (6.33), then by Lemma 5.6 we deduce that u> uc for any c > 0. Therefore u > u. To show the existence of a maximal solution for (6.33), we consider the problem -Au

= au-b(x)up

in ftn, u|an„\r 0 = °°; u lr 0 = 0,

(6.37)

where !1„ E { i e ( 1 ; d(x) > 1/n}. By what has just been proved, this problem has a minimal positive solution un. Using Lemma 5.6, we see that for any positive solution u of (6.33), un > u, un > M„_|_i in fln. Hence u(x) := l i m n - ^ un(x) is welldefined and u(x) > u(x) in Cl. But one easily sees that u is a positive solution of (6.33). Hence it is a maximal positive solution. This finishes the proof of Step 1. S t e p 2: The general case. For each positive integer n we define bn(x) — mm{b(x), n} and consider (6.36) with b(x) replaced by bn{x) and denote its unique solution by ucn. By Lemma 5.6, we know that ucn is nondecreasing in c and non-increasing in n. Therefore uc(x) — linin-KjoU^a;) is well-defined in ft, and 0 < uc{x) < u^(x) in ft. It follows that uc(x) —» 0 as d(x,T0) —» 0. Moreover, by standard interior Lp estimates, uc is C 1 in ft and satisfies (6.34). We claim that uc(x) —> c as d(x) —> 0. To prove this claim, we let <j>n be the unique solution of A(j> = 1 in ns, 0|rTO = c, (Alan^rco = <|anAr=c>

Boundary Blow-Up

Problems

105

where 5 > 0 is a small constant and fi<s := {x £ fl : d{x) < 6}. Then wn := ucn — (j>n satisfies, - A w n = a{x)ucn - bn(x){ucn)p + 1 := fn in Q.s, wn\dns and since \fn\ < ||o||ooUi + c0d(x)a(ul)p

= 0,

+ 1, and a > - 2 , we have

d(x,dnsy\fn(x)\

for some 7 G (1,2) and C > 0 independent of n. Therefore we can use Lemma 4.9 (and Problem 4.6) in [Gilbarg-Trudinger] to conclude that

d(x,dnsy-2\wn(x)\ < d in ns, which implies that \uc{x) — c\ -+ 0 as d(x) -> 0. Therefore u c is a solution to (6.36). The rest of the proof is now exactly the same as in Step 1, namely, the minimal solution is u := lim^oo uc, and the maximal solution is the limit of the minimal solution of (6.37) as n —+ 00, except that to see u(x) —> 0 as d(x, To) —* 0, we use 0 < uc < ucn and hence 0 oo

By Step 1, un solves (6.33) with b(x) replaced by bn(x), in particular un(x) —> 0 as d(x, To) —* 0. By the above inequality, the same limit holds for u. D Theorem 6.15 that

Suppose there exist constants a > — 2 and c > 0 such

,

hm d{x)^o

Kx) = c. a d(x)

Then problem (6.33) has a unique positive solution u and di™ 0 d(x)-(-+2)/( P -D Proof.

_

L

c(p- l)^

J

•

(b 36j

-

For any given small e > 0 , w e f i x a 5 > 0 small such that

(i) d(x) is C 2 for x satisfying 0 < d(x) < 26; (ii)

-Ad(x)s

— a(x)s2

< e for s € [0, 2<5] and x satisfying 0 <

d(x) < 25; (iii) (c - e)d{x)a < b{x) < (c + e)d(x) Q for a; satisfying 0 < d(x) < 25.

106

Maximum

Denote (3 = - ^ f ,

Principles

and

and let

(/3(/?;+1^-e)1/(P'1), ? = (

i =

Applications

^

+ i

)

1 / M

,

(6.39)

and for cr € (0,5), define ^ = [d{x) + aft,

va = [d(x) -

afl

We first consider the case a > 0. Since |W(a:)| = 1, and b(x) > (c — e)[d{x) — a\a when cr < d(x) < 25, we easily obtain - a(x)vcr + b(x)(va)p

-Ava

= f { - (3[d{x) - af~xAd{x) -a(x)[d(x) > £[d(x) - af~2{ > 0

- /3(/3 - 1) [<*(*) -

- of + b(x)[d(x) -

af~2

afrf1}

- f3Ad(x)[d(x) - cr] - a{x)[d{x) - a}2 + e l

for all x satisfying a < d(x) < 26.

Similarly, since b(x) < (c + e)[d(x) + a]a when d(x) + a < 25,

-Awff-fl(iK + ^ ) ( ^ ) p < §[d(x) + (rf~2{ < 0

- pAd{x)[d{x) +a}-

a(x)[d(x) + a}2 - e l

for all x satisfying d(x) + a < 25.

Denote £1$ = {x S fi : d(x) < 5} and consider the problem - A w = a(x)w - b(x)wp in Q6, u/| roo = 0, w | a Q A r o o = 00. By Theorem 6.14, this problem has a minimal solution w. Suppose that u is any positive solution of (6.33). Then one easily checks that v = u + w satisfies —Av > a(x)v — b(x)vp

in Qg.

Since v

\rx

= 00 > ^ | r o o ,

and w

lanAr=o = °° > ^ l a n A r ^ .

by Lemma 5.6, u + w^v^

onfls.

(6.40)

Boundary

Blow-Up Problems

107

Similarly, for a £ (0, S), va + w > u on

fii\nCT.

(6-41)

Letting a —> 0 in (6.40) and (6.41), we deduce d(xf£

+ 2w>u

+ w> d(xf£,

Vx e 0,$.

It follows that i <

ljmd{x)^0d(x)-'3u(x)

< K^d^odixyPuix)

(6.42)

_ f(2+ a )(l+a+p)\ 1 / ( p - 1 )

(6.43)

Due to (6.39), letting e —> 0 in (6.42) we obtain \imd{x)^0d(x)-Pu(x)

=

( ^ i )

V(p-i)

Suppose now wi and u2 are two positive solutions of (6.33). By (6.43), for any e > 0, lim \u\(x) — (1 + e)u2(x)] = —oo, lim [1x2(2;) — (1 + e)ui(x)] = — co. d(x)—>0

d(x)—>0

Let us denote u)j = (1 + e)uj, i = 1,2. Clearly —Auii > awi — b{x)w? in fi, Wi = 0 on ToHence we can use Lemma 5.6 to conclude that ui(x) < (1 + e)u2(x),

u2(x\<

(1 + e)ui(x),

\/x G fi.

Letting e —> 0, we obtain Mi = u2. This finishes the proof of the theorem for the case a > 0. It remains to consider the case a S (—2,0), where the above arguments do not work. Let 6 > 0 be small enough such that (i)-(iii) at the beginning of the proof hold and d(x, dfi) = d(x, Too) in Q2s- Then by our assumption, there exist c2 > c\ > 0 such that cid(x)Q < b(x) < c2d(x)a

Vx e fi26.

It follows that any positive solution u of (6.33) satisfies —An < ||a||ooU — cid(x)aup

in £l2g.

Maximum Principles and

108

Applications

For any given XQ € ilg, consider the ball BQ = {x € RN d(x0)/2}. In B0, d(x) < (3/2)d(x 0 ) and hence - A u < Holloou -

a p

Cl[(3/2)d(xo)}

u

: \x — XQ\ <

in Bo-

Denote C = (l/2)d(x 0 ) and define, for \x\ < 1, v(x) = C{a+2)/(p~1)u(x0

+

£x). Then -Av

3aClvp in Bi(0),

where A — ||a|| 0 0 [(3/2)max a : e n^(^,5n)] . Let VQ be the unique solution to -Av

= Av - 3acivp in 2?i(0), v\9Bl(0)

= oo.

By Lemma 5.6 we deduce v < VQ in JE?I(0). In particular, u(0) < vo(0), i.e., u(x0) < r ( a + 2 ) / ( p _ 1 ) w o ( 0 ) =: c 0 d(a;o)- ( a + 2 ) / ( ''- 1 ) , Vx0 e fl*. Define w(x) by w = w

1

with 7 > 0 to be determined later; we obtain

_ A w + ( 7 + 1} LYH! = _ % + M^d-rt+i. w

7

7

We observe that tu is bounded over 0,$ and wlr^ = 0. Let v\ be the unique solution of -Av

= 0 in ils, v\Toa = 0, u| d ( l)=<5 = w| d(x)=(5 ,

and let wi be the unique solution of - A w = / in ils, w\ene = 0, where f = -&w+

*MW7(I-P)+I.

Then

- A ( i u i + ui) = / , - A w < / in Qs,

(wi + V1)\9Q5 = VJ\9Q6.

By the maximum principle we deduce w < wi + v\ in fij. On the other hand, if we have chosen 7 > 0 large enough such that 7(1 -p) + 1 < 0, then, using u(x) < c0d{x)-(a+2V(p-1'> in Sls, which we have obtained earlier, we deduce b(x)w«1-ri+1

It follows that 1/(1)1 ^ c i ^ . a f i , ) - 2 4 " ^ ? ^ .

Boundary

Blow-Up

Problems

109

We may apply Lemma 4.9 (and Problem 4.9) in [Gilbarg-Trudinger] to conclude that wi < C2d(x,dtts)-' p- 1 , Vx e QsSince v\ 6 L°°(Qs): this implies that 1_ Q + 2

w < Wi + vi < czd(x) i p- 1 , Vx G fi<j/2. Therefore u(x) > Cid(x)~(-a+2^(-p~1^ estimates, we know that c4 < u(x)d(x)ia+2)/ip-V

in fi<s/2- Combined with our earlier

< c0 in Qs/2

for any positive solution u of (6.33), where the positive constant £4 may depend on u. Suppose now u and u are the minimal and maximal positive solutions of (6.33). Then the above estimate implies u 1. If To is not empty, then we can apply the Hopf boundary lemma to see that the above inequalities also hold near To if we enlarge c when necessary. Therefore, by possibly further enlarging c, we may assume that the above inequalities hold in tt. If u^.u, then a simple application of the strong maximum principle shows that u v > ——u, -v + -u = u. ~ 2c 2c + 1 2c+1 Denote f(x,t) = -a(x)t + b{x)V>. We find that §^f(x,t) > 0 for x € Q and £ > 0. Therefore f(x,t) is convex in £ for £ > 0 and thus 2c

1

_

It follows that -Av

= - ( 1 + — )/(z,w) + ^ / ( x , u ) >

that is, —Aw > a(x)v — b(x)vp in fi.

-f(x,v),

Maximum

110

Principles

and

Applications

We can now apply Lemma 5.6 to compare v with the unique positive solution uc defined in Step 2 of the proof of Theorem 6.14, and obtain uc < v, which implies u < v. But by definition, v < u. This contradiction shows that we must have u = u, that is, (6.33) has a unique positive solution. Let us observe that the same argument also shows that the problem -Aw = a(x)u - b(x)up in 0,s, w| roo = oo, u\d{x)=s

= 4>

has a unique positive solution, where > 0 is continuous. We are now able to obtain the blow-up estimate near Too. Let e,S,£,£ and 21CJ, va be defined as at the beginning of the proof. We now take a = 0 and find that v0 and vo satisfy -AiJo > av0 — bvQ, —A^o < av^ — by^ in Q.2&. Since b(x) > cid(x)a > c$ > 0 in Q.26 for some constant c$ > 0, we easily see that for all large t, say t > To, a(x)t — b(x)tp is decreasing in t and is negative for every x £ H.2S- For any given positive solution u of (6.33), we now take $1 £ (0,2(5) small enough such that y_0(x),v0(x),u(x) > 2TQ in Qg1. Denote Mi = min u(x), M2 = max d(x)=5\

u{x).

d(x)=Si

Then it is easily checked that 61 -^Ho -niax{w 0 (Ji) - M i , 0 } , v2 :=v0 + max{M 2 -

v0(5i),0}

are lower and upper solutions of the problem -Av

= av- bvp when a < d(x) < Si, v\d{x)=Sl

= u, v\d{x)=cr = v 0 ,

where 0 < a < Si. Since vi 2 in flg1, we conclude that the above problem has a minimal solution ua satisfying vi < ua < v2- Moreover, u„ is non-increasing in a. Hence UQ := lmv_,o ui < UQ < v2 in £1$! and - A u 0 = au0 - bup0 in 9,&l, u0\roo = 00, u\d(x)=Sl

= u.

By our earlier discussion, we know that this problem has a unique positive solution. As u is a positive solution of this problem, we must have UQ = u, and hence vi 0

= £,

Boundary

lim^xj^o^)

Blow-Up

0

u(x)>

Problems

lim d{x)

111

'3vi(x)=^.

Letting e —> 0, we obtain (6.38). The proof is now complete.

•

We next demonstrate that the rate of blow-up is local in nature. To this end, we assume that B is a small open ball satisfying B n dQ, ^ 0 and consider the problem —Au — a(x)u — b(x)up in ft n B, ulanns = oo,

(6.44)

where, as in (6.33), a(:r) is a continuous function in fl and b{x) is continuous and positive in fl, p > 1 is a constant. Theorem 6.16 Suppose that dfl is C2 and 2* € 9fi n B. (i) If for some constants a > —2 onii c„ > 0, E

"— e f i d(^F- C *'

(6 45)

-

t/ien evert/ solution u of (6.44) satisfies u(x) x x xen - -* -' d(x,dCL)-(<x+2)/(P-i)

[(2 + a ) ( l + Q + p ) i 1 / ( p - i ) ~ I c„(p-l)2

(6.46)

(ii) If for some constants a > — 2 and c, > 0, 1M

— -^d^F^ 0 "

(6 47)

'

£/ien even/ solution u of (6.44) satisfies u{x) r(2 + a ) ( l + a + p)i 1 /(p-i) d(l,an)-( + )/(P- ) - L C,(p-1)2

limr^x^.TCQ—

—— a; 2-^TT, 17T <

(6.48)

Proof. Let e > 0 be fixed. By (6.45) we can find some ro > 0 small enough such that Bro(x.)

:= {x € RN : \x -

Xt\

< r0} C B,

(6.49)

and 0 < 6 ( x ) < (c*+e)d{x,dfl)a,

Vi6Bro(i,)nn.

Let 0 i and 02 be two smooth domains such that 0\ CC 0 2 CC Bra(x*) and O i n fl = 7, C <9fl with x , belonging to the interior of J«, namely, 0\ is outside fl but its boundary and dfl has a common part i»

Maximum

112

Principles

and

Applications

which contains x* in its interior. Set D* = 02 \ ®\ continuous function over D*. By Theorem 6.15, the boundary value problem f -Au

= a(x)u-(c* V;

I

+ e)d(x,d01)aup

aj

id extend a(x) to a

in £>,

(6.50)

|u = 0 on<90 2 subject to u = n > 1 (resp., u = oo) on dO\ has a unique positive solution wn (resp., W). Moreover, we have W(x) lim d(x,ao1)^od(x,d01)-(a+2)/(p-11

r(2 + a ) ( l + a + p) L (c, + e ) ( p - l ) 2

i/(p-i)

(6.51)

Since rf(x, <9f2) for x 6 D* n fi, we have, for all n > 1 and any solution u of (6.44),

(

— Aw n < a(x)wn — b(x)wpl

in D* n f2,

iun = 0 on cXt?2,

"Ic.nan = °° > ^n ID.nan-

Using Lemma 5.6, we infer that ( wn< wn+1 < W in £>», Vn > 1, 1.

(6.52)

Standard regularity arguments show that w^ defined by woc(x) := limn-,00 wn(x), \fx E D», is a positive solution of (6.50) satisfying w ^ = oo on dO\. It follows that w^ = W in D*. By (6.52), we obtain w > W in D* n fi. Hence, u(x) Q 2

1

> -77—

Wlx) ;a ,2„ w , 1, , ,

d(x,5n)-( + )/(p- ) - d(x,sn)-( + )/(p- )'

VxEA,nfL

By our choice of 0\, d{x,dO\) = d(x, dQ.) if x £ fHs sufficiently close to x». Hence letting x —> x* and using (6.51), we find u(s) -x-+x„xend^x^dViy{a+2)/{p-i)

^ r(2 + Q)(l + q + p ) - | i / ( p - i ) - [ (c
Since e > 0 is arbitrary, we obtain (6.46) by letting e —> 0. This proves (i). To show (ii), for given small e > 0, we choose ro > 0 small such that (6.49) holds and b{x) > (c*-e)d(x,dn)a

Vx e 5 r o (x») n SI.

Boundary

Blow-Up

Problems

113

Let V be a C 2 domain such that V C O n B r o ( i , ) and I* := <9V n <9fi contains x* in its interior. For n > 1, set Vn = {x G V : 1/n < ^(z.dV)}. Obviously, V = U°° , V u n = l K "-

Let Z be the unique positive solution of - A u = a(x)u - (c* - e)d(x, dV)aup

in V, u\dV = oo.

(6.53)

Let Zn be the unique positive solution of the above problem with V replaced by Vn. By Theorem 6.15 we have Z{x) _ r(2 + Q)(l + g + p)-[ 1 /(p- 1 ) d(x,rHorf(x,9y)-(«+2)/b-i)"i (Ci|i_c)(p_i)2 . Clearly d(x, 5V„) < d(x,dV„ + i) < d(x,dfl) f - AZ„ > a(x)Zn - b(x)Z*

(6.54)

for each x G Vn, and in Vn,

[ Zn\dvn = oo > u|av n ,

Vn > 1, Vn > 1.

By Lemma 5.6, we deduce that j Zn>

Zn+i

( Zn > u

in Vn, in Vn,

Vn > 1, Vn > 1.

(6.55)

For each x G V, there exists an integer m(x) > 1 such that x G Vn for n > m{x). By virtue of (6.55), Zoc(x) = \imn->oo Zn(x) is well defined and Zoo > w. Standard regularity arguments imply that Zoo is a positive solution of (6.53). Since there is only one such solution we conclude that Zoo = Z in V. It follows that

d(x,dfr)-(Q+2)/(p-1) - d(x,afi)-(Q+2)/(p-1) Since
u(x) r(2 + a ) ( l + a + p ) i 1 / ( p - i ) 1 d(x,afi)-( Q + 2 )/(p- 1 ) " L ( c * - e ) ( p - l ) 2

Passing to the limit e —* 0, we arrive at (6.48). This finishes the proof.

•

Using Theorems 6.15, 6.16 and Lemma 5.6, we immediately obtain the following result.

Maximum

114

Principles

and

Applications

Corollary 6.17 Suppose that dQ, is C2 and there exist constants C2 > ci > 0 and a > — 2 such that a

< b(x) < c2d(x)a

Cld(x)

in

ft.

(6.56)

Then any positive solution of (6.33) satisfies

°i * d(g) -("S)/(p-i) ± < * » f i '

(6-5?)

where C\, C2 are positive constants independent of u. We can prove that condition (6.56) implies uniqueness for (6.33) though it does not give an exact blow-up rate for the solution; this improves the uniqueness part of Theorem 6.15. Theorem 6.18 Under the conditions of Corollary 6.17, (6.33) has a unique positive solution. Proof. By Theorem 6.14 we know that (6.33) has a minimal positive solution u*. Arguing indirectly, we assume that (6.33) has a positive solution u such that u ^ u*. By the strong maximum principle, we easily deduce that u > «* in Q. Due to (6.57) and the Hopf boundary lemma (in case To 7^ 0), we can find a constant k > 1 such that u < /CM* in ft. Define v = u, — (2k)~1(u — u»). Clearly u

^

> v

>

k +1

Ik

_ „ „

_ _ „

1 +

_ _ _

u

=

„..

^

^

Denote f(x,t) = - a ( x ) i + 6(x)iP. We find that -j£rf(x,t) > 0 for x G ft and £ > 0. Therefore / ( x , £) is convex in t for £ > 0, and by (6.58),

/(

u

)

2fc

*' * - 2 rn

/M +

1

2Tn /(x ' u) -

It follows that - A t ; = - ( 1 + ^)f(x,

u.) + i / ( a : , u) > - / ( x , v),

that is, —Aw > a(x)v — b(x)vp in ft.

Boundary

Blow-Up

Problems

115

We now apply Lemma 5.6 to compare v with the unique positive solution uc of (6.36) and find v > uc in Q. Letting c —> oo, we obtain v > u* in fi, which contradicts (6.58). • Remark 6.19 (i) Theorem 6.15 with a > 0 seems first proved in [DuHuang(1999)], which was extended to the p-Laplacian case in [Du-Guo(2003)]. Other generalizations and related results can be found in [Cirstea-Radulescu(2002)[, [CirsteaRadulescu(2004)], [Garci'a-Letelier-Sabina de Lis(2001)], [LdpezG6mez(2003)], [Zhang(2000)]. (ii) For the case a > — 2 but a(x) = 0, Corollary 6.17 and Theorem 6.18 were proved in [Chuaqui et a/.(2004)], see also [Chuaqui et «L(2003)] for further related results. (hi) Theorem 6.18 with a > 0 can be found in [Du(2004b)], which was extended in [Cirstea-Du(2005)] to more general nonlinearities but with bounded weight functions. In both [Du(2004b)] and [CirsteaDu(2005)], a version of the so called "Safonov iteration technique" was used in the uniqueness proof, which is different and more involved than the one given above. (iv) The convex function trick used in the proof of Theorem 6.18 was first used by Marcus and Veron in [Marcus-Veron(1998a)] (see also [Marcus-Veron(2003b)]). (v) The existence of multiple boundary blow-up solutions was considered in [McKenna-Reichel-Walter(1997)], [Aftalion-Reichel(1997)], [Aftalion et a/.(2003)], [Du-Yan(2004)J, [Du-Guo(2006a)] and [DuGuo-Zhou]. (vi) Theorem 6.16 can be easily extended to the p-Laplacian case; we leave this to the interested reader. Remark 6.20 If in (6.33) we replace ulr^ = oo by wlr^ = Mi where \i is a (possibly unbounded) regular Borel measure on dCl, the problem then is known as a "boundary trace problem". Such problems arise from "superdiffusion" processes in probability theory, and were systematically studied by probabilistic methods (see [Dynkin(2002)]) and analytic methods (see [Marcus-Veron(1998a); Marcus-Veron(1998b); Marcus-Veron(2003a)]). Remark 6.21 If in (6.33) we replace fl by a compact Riemannian manifold M (of dimension N > 3 with metric g) and understand A as the Laplace-Beltrami operator A s , and let Too be a nonempty closed subset

116

Maximum

Principles

and

Applications

of M, then the problem (with p = ^ j § ) becomes a "singular Yamabe problem" in geometry. One is interested to know what kind of Too is admissible and for admissible Too what is the asymptotic behavior of u near Too. We refer the interested reader to [Loewner-Nirenberg(1974)], [KazdanWarner(1975)], [Finn(2000)] and [Fakhi(2003)] for further details.

Chapter 7

Symmetry and Liouville Type Results over Half and Entire Spaces In this chapter, we look at some results on the symmetry of solutions which are different to those considered in Chapter 3, where the "moving plane method" was used. Here our arguments are again based on the maximum principle, but in many cases, the symmetry or partial symmetry of the solution comes from some kind of uniqueness results.

7.1

Symmetry in a half space without strong maximum principle

In this section, we are concerned with positive solutions of elliptic equations of the type -Au = f{u)mTcRN{N>2),

u = 0ondT,

(7.1)

where T = {x = (xi, x 2 , . . . , XN) : xi > 0}. If / is C 1 and satisfies (i) /(0) > 0 or /(0) = 0 and /'(0) > 0, (ii) for some a > 0, f(u) > 0 for 0 a, (iii) f'(a) < 0, then a well-known result of Angenent [Angenent(1985)] says that any bounded positive solution of (7.1) is a function of x\ only. The proof in [Angenent(1985)J relies on the strong maximum principle, which is applicable to (7.1) because / is C1. We will use a different method to show that this result remains valid even if the strong maximum principle cannot be 117

Maximum

118

Principles

and

Applications

used as in [Angenent(1985)]. This allows us to treat cases where / is less smooth than locally Lipschitz continuous and hence the case where the solution may have a "flat core", namely, a solution u with {x GT : u(x) = a} nonempty (which is usually called a flat core). This kind of equations arise from various models in chemical reaction theory and population biology. We say that f(s) is locally quasi-monotone on [0, oo) if for any bounded interval [s\, $2] C [0, 00), there exists a continuous increasing function L(s) such that f(s) + L(s) is non-decreasing in s for s € [s\, S2]. This condition is less restrictive than requiring f(s) to be locally Lipschitz continuous on [0,oo). We will show that for a broad class of locally quasi-monotone nonlinearities f(u) satisfying f(u) > 0 in (0, a) and f(u) < 0 in (a, 00), any positive solution u(x) of (7.1) is a function of x\ only, i.e., u(x) = and y(t) is a positive solution of the problem -y" = f(y)

in (0,00),

2/(0) = 0.

y(xi), (7.2)

By a positive solution of (7.1) we mean a function u e C 1 (T) satisfying u > 0 in T, u = 0 on dT and f Du • Dipdx = [ f(u)i>dx, V?/> e JT

C^{T).

JT

In the following, we first list our main results of this section, and then we prove them one by one by introducing suitable techniques. Theorem 7.1 Let f{s) be continuous and locally quasi-monotone [0,00) and satisfy the following conditions: (Fi) For some a > 0,

on

f(s) > 0 in (0,a), f(s) < 0 in (a, 00). (F2) There exist constants 5 > 0 and a > 0 such that f(s)>5s,

VsG(0,a).

Then any bounded positive solution u(x) of (7.1) satisfies u < a on T and depends only on the variable xx; hence u(x) = y(xi) in T and y(t) is a positive solution of (7.2). It will be proved later that (7.2) has a unique positive solution. Though the strong maximum principle fails in general under our assumptions here, but since f{u) is nonnegative for small positive u, we can

Symmetry

and Liouville Type Results over Half and Entire Spaces

119

apply Theorem 3.9 (with some (3{u) satisfying (3(u) = 0 for all small u > 0) to conclude that any nonnegative solution of (7.1) is either identically 0 or strictly positive in T. Therefore, under condition (Fi), any nontrivial nonnegative solution is strictly positive in T. By making use of boundary blow-up solutions, we can easily prove the following result. Theorem 7.2 Suppose that f is continuous and locally quasi-monotone on [0,oo). Then any positive solution of (7.1) is bounded provided that f satisfies further {F3) for some large M > 0, there exists a continuous function g(s) such that f(s)
VsG[M,oo), fU r » r c» -, -1/2 du < 00. / \g(s)\ds

/>0O

g(s) is nonincreasing in [M, 00) and /

JM

*-JM

Depending on the behavior of f(u) near u = a, a positive solution u of (7.1) may reach a at some points x £ T. In such a case, the set O := {x £ T : u(x) — a} is called the flat core of u. We have the following result. Theorem 7.3 that

Under the assumptions of Theorem 7.1, suppose further

(F4)

[a[F(a)~F(s)]-1/2ds< Jo

Then the unique positive solution of Theorem 7.1 has a flat core, and it is given by 0 = {x£T:

Xi>

t0},

where to = (l/y/2)

fa[F(a)-F(s)}-^2ds Jo

and F(s) = [' Jo

Conversely, if (F4) is not satisfied, that is, if

(F 5 )

[a[F(a)-F(s)]-1'2ds Jo

= 00,

f(£)d£.

120

Maximum

Principles

then the unique positive solution and hence u < a onT.

and

Applications

of Theorem

7.1 does not have a flat core,

An illustrative example for / is the following one, = u(a - u)\a - u\q~l,

f0(u)

Mu > 0,

where q > 0. It is easily checked t h a t /o is quasi-monotone and satisfies (F\) — (Fz). Moreover, it satisfies (F5) if q > 1, and it satisfies (F4) if 0) n C(V) and satisfy in the weak sense, for some e > 0, - A u > h(u),

-Avt

u>vt

< h(vt) -e

inV,Vt

£

[ti,t2],

+ e on dV, Vi £ [ti, t2\.

Moreover, suppose that u > vto in V for some to £ [ti, £2] and t —> vt is continuous from the compact interval [^1,^2] to C(T>). Then u>vt

onV,

Vt£

[h,t2)-

Proof. Denote T = {t £ [t 1; t2] : u > vt on V}. Clearly t0 £ T and T is a closed set. We show t h a t T is relatively open in [^1,^2], which would imply T = [ti,t2], as required. Since vt varies continuously with t, it is easily seen t h a t there exist finite numbers s\ < s2 such t h a t u(x),vt(x) £ [s\,S2] for all x £ V and all t£ [*i,t 2 ]. Since h(s) is locally quasi-monotone in s, we can find a continuous increasing function L(s) such t h a t h(s) := h(s) + L(s) is nondecreasing in s for all x £ V and s £ [s\, $2]. Let 5 > 0 be sufficiently small. Then, for any t £T, - A w + L(u) > h(u) > h(vt) >-A{vt

> -Avt

+ L(vt) + e

+ S) + L{vt + 5) in£>,

Symmetry

and Liouville Type Results over Half and Entire Spaces

121

and u > vt + S on dV. We claim that u > vt+6 in V. Otherwise, D» := {x G V : u(x) < vt(x)+5} is nonempty and on each component Do of D*, we have, for w = vt + 5 — u, -Aw

< L(u) - L{vt + S) < 0 in D 0 , w\dDo = 0.

Therefore we can use the weak maximum principle to conclude that w < 0 in D 0 , which is a contradiction. It follows from this claim that for all s G [£1,^2] with \s — i| small, u > vs. This shows that T is relatively open in [*i, ^2]- The proof is complete. • We now make use of the weak sweeping principle to obtain some bounds for positive solutions of (7.1). In the following we always assume that f(s) is continuous and locally quasi-monotone on [0,oo). Proposition 7.5 Let (Fi) be satisfied and u be any bounded positive solution of (7.1). Then u < a. Proof. Let M > 0 be such that u{x) < M on T. We may assume that M > a for otherwise there is nothing to prove. Let •0° be the unique positive solution of -AV>° = 1 i n B ^ O ) ,

V(Wo)=0.

Then ip° is radially symmetric and ^"(O) = HV^Iloo- F° r any give 77 > 0, by 6r, := -

max

f{s) > 0.

(7.3)

For arbitrary XQ G T, t > 0 and A > 0, we define 1>t,x(x) =M + 7]-tip°(\-1{x-xQ)),

xeBx{x0).

Let £ i = 0 and t2 = (M - a)/0(0). Clearly, for t €

[h,t2],

a + v < ipt,\(x) < M + r), Vx G Bx(x0) and -Aipt,\

= -t/X2

In view of (7.3), we have, for A >

> -t2/X2,

\/x G

Bx(x0).

(2/Sr))1^t12/2,

-AV't.A > /(VvO + V 2 > Vx G Bx(x0)yt

G [t!,t2].

(7.4)

Maximum

122

Principles and

Applications

Clearly, iptltx(x) = M + ri>u{x)

\/x£Bx{x0)nT1_

ipt}X{x) = M + t)>«i(,ViedBx(xo)n T, Vt e [ti,t2], ipt,\{x) >a + r]>u + r) VxeBx(x0)ndT,Vte

(7.5)

[h,t2].

Thus, due to (7.4) and (7.5), we can apply Lemma 7.4 to conclude that u < tpt,x on Bx(x0) f l T for all t € {h,h] provided that A > {2/5r))1'2t\/2. In particular, U(XQ) < ipt2:x{xo) = a + r\. Since xo 6 T and 77 > 0 are arbitrary, this implies that u(x) < a on T. • Proof of Theorem 7.2. Define h(s) = — g(s) for s > M and /i(s) = —g(M) for s < M. Then /i(s) satisfies all the conditions of Corollary 6.3. Therefore, we can find R > 0 small enough such that the problem -Au

= -h(u)

in BR(0),

U\9BR(O)

= 00

has a minimal solution v and u(x) > v(0) > M in -Bfl(O); this last property follows from (6.5) and the definition of g{t). Note that v e C1(BR(0)) by standard regularity theory. Let u be a positive solution of (7.1). We want to show that u(x) < v(0) in T. Otherwise, we can find xo € T such that u(x0) > v(0). Letting v0(x) = v(x — xo), we find vo S C1(BR(X0)), -Av0

=-h(v0)=

g(v0) inB f l (xo),

v0\dBR^Xo) = 00,

and U(XQ) > v0(x0). Let V be a component of the set { x e T f l BR(X0) u(x) > v0(x)}. We find u(x) > v0(x) in V and w = i>o on dV. Thus,

:

—A(u — v0) = f(u) — g(v0) < g(u) - g(vo) < 0 in V, u — vo — 0 on dV. By the weak maximum principle, we deduce u < vo in T>. This contradiction completes the proof. • Next we provide two crucial steps towards a complete proof of Theorem 7.1. Lemma 7.6 Suppose that f satisfies (Fi) and (F2). Then problem (7.1) has a minimal positive solution u among all bounded positive solutions. Proof. Let ^ be a positive first eigenfunction for —A over BQ under Dirichlet boundary conditions, i.e., -A<£ = Ai(B o )0 in Bo, 3RQ. We can choose R0 sufficiently

Symmetry

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123

large so that \\(Bo) < 5/2, where 5 is given in (F2). We extend <j> by 0 on RN\BQ and denote it by <j>. Then for all small positive constant e and any R > 3Ro, it is easily checked that e(f> is a weak lower solution of the problem -Au

= f(u)

in TR,

U\9TR = 0,

(QR)

and € < a. Clearly a is an upper solution of {QR). Thus (Q.R) has a minimal positive solution UR in the order interval [e, a] := {u G W 0 ' (TR) : £ R, we can also obtain a minimal positive solution URX in the order interval [e, a]. We claim that Will > UR in TR. Indeed, we know that UR1 \TR is an upper solution of (QR) and UR1 > e<j> in TR. Thus, (QR) has a positive solution in the order interval [e^, URJ] C [e<j), a]. Since UR is the minimal positive solution of (QR) in the order interval [e<^,a], we must have UR < UR1 in TR. This proves our claim. Therefore, if we choose an increasing sequence {Rn} with Rn > 2>RQ for all n and Rn —> 00 as n —> 00, and denote un := «Rn and T n := TRn, then we have a > w n +i > u n in T n , Vn > 1. Therefore u(x) = lim n _ 0 O ^ ( a ; ) is well-defined in T and by a standard regularity argument, it is a nontrivial nonnegative solution of (7.1). By our earlier observation (before the statement of Theorem 7.2), it is a positive solution of (7.1). We will show that if e has been chosen small enough, then u is the minimal bounded positive solution of (7.1). To this end, we make use of the weak sweeping principle again. Let BQ and <j> be as before. We first determine a constant £2 > 0 by ^H^lloo = c/2. For an arbitrary bounded positive solution u of (7.1), we can find c\ > 0 such that u > e\ on BQ. Then choose t\ e (0, t2) such that t\ |||oo < £i, and for 77 > 0 small, denote by BQ t n e concentric ball of BQ with radius R0 — 77. We can find 770 > 0 small enough such that t2\gBt < ej/2 for all 77 e (0,770]- We now define vt = t(f> for t G [*i,*2], and obtain - A u t = Xi(B0)vt

< (S/2)vt < (l/2)f(vt)

< f(vt) - e, in B%,

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and vt 0 and all 77 € (0, 770] and all t e [t1; £2]- By our choice of t\, we have u > vtl on BQ. Therefore we can apply Lemma 7.4 to conclude that u > vt2 = t2<j> in BQ. Since 77 e (0,770] is arbitrary, it follows that u > t2 in BoThis implies that if we have chosen e < t2, then u > ecf> in T. By Proposition 7.5, u < a. Moreover, u satisfies - A u = f(u),

u\dBRn

> 0.

Thus, u is an upper solution of (Qi? n ). As we have u > e, (QRn) has a positive solution in the order interval [e,u] C [e,a]. This implies that ?j > un in T n for each n, and hence M

> u = lim u n . n—»oo

So u is the minimal positive solution of (7.1).

•

Lemma 7.7 Suppose that f satisfies (Fi) and (F2). T/ien problem (7.1) has a maximal positive solution u among all bounded positive solutions. In order to prove this lemma, we first prove the following result. Lemma 7.8 Let R > 0, TR = BR(0) nT,TR = BR(0) n dT and HR = dBR(0) (~)T. Suppose that f(s) satisfies (F\) and (F3). Then the problem - A u = f(u)

in TR,

u\rR = 0,

U\HR

= 00

(7.6)

has a positive solution. Proof.

Let p(x) = p{x\) = x\. We consider the problem - A u = /(u) in TR,

u\dTR = p.

(7.7)

Clearly 0 is a lower solution and the constant max{a,maxy p} — max{a,i?} is an upper solution of (7.7). Therefore there exists a maximal positive solution uR which satisfies 0 < UR < max{a, R}. For any integer n > 1, we now consider the problem -Au

= f(u)

in TR,

U\9TR = np.

(7.8)

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125

By the above argument we know that (7.8) has a maximal positive solution un in the order interval [0,max{a,ni?}]. Now we show that for all large n, «n+i > un in TR. In fact, it is obvious that un is a lower solution of the problem -Au

= /(«)

in TR,

U\9TR = (n + l)p,

(7.9)

and max{a, (n + l)R} is an upper solution to this problem. Since un < max{a, nR}, we find that maxw n < (n + l)it! for all large n, and hence the lower and upper solution argument implies that (7.9) has a positive solution in the order interval [un, (n + 1)R]. It follows that un < un+i in TR. The argument in the proof of Theorem 7.2 can be easily used to show that {un{x)} is bounded from above for x in compact subsets of TR U TR. Hence, uR{x) := linin-xx, un(x) exists. A standard regularity consideration shows that uR is a solution of (7.6). • Proof of Lemma 7.7. By suitably modifying f(u) for u > a, we may assume that / also satisfies (F3). Now we choose an increasing sequence of positive numbers {Rn} such that R„ —> 00 as n —> 00 and denote Tn = TR„ . By Lemma 7.7, for each n, the problem (7.6) has a positive solution M" on Tn, and for later use, we assume that this solution is constructed as in the proof there. We claim that un > un+1

in T„.

Indeed, by standard regularity theory we know that un+1 is C 1 in Tn and hence there exists an integer K > 1 such that un+1 < Kp on T n , m&xKp = KRn > a. Thus, M n+1 is a lower solution of the problem - A M = f{u) in Tn,

u|sr n = Kp.

Since KRn is an upper solution of this problem, we find that un+1 < u^ in Tn, where UK is the maximal solution of this problem in the order interval

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[0,KRn\. By the construction of un, we find w™ > UK- Therefore un+1 < un inTn. From our construction of u in Lemma 7.6, we easily see that un > n+l u > u for all n. Hence u = lim un > u n—>oo

is well-defined on T. Furthermore, by standard regularity considerations, we know u satisfies (7.1) on T. Clearly any bounded positive solution u of (7.1) satisfies, for each n, - A u = f(u),

u\dTn < Kp,

for some K > 1. It follows that u < un on T„, and hence u = lim un > u. n—»oo

Note that by Proposition 7.5 and Theorem 7.2, u < a in T. Therefore u is the maximal bounded positive solution of (7.1) with the original unmodified / . The proof is now complete. • We need one more ingredient for the proof of Theorem 7.1, which is about the following ordinary differential equation problem -w" = f(w), 0 < w < a i n (0,oo),

w(0) = 0.

(7.10)

Lemma 7.9 Suppose that f satisfies (Fi). Then (7.10) has a unique solution w(t). Moreover, w'(t) > 0 and w(t) —> a as t —> oo. Proof. We first assume such a solution exists and find its properties. Once we know the properties, the solution can be determined explicitly. We show that w'(t) > 0 for t G [0, oo). The first integral of the equation gives that 1

-\w>(t)\* + F(W(t))=1-\w'(0)\\

(7.11)

where F(w) = f™ f(£)d£. Since F(w) > 0 for w e (0,a], we necessarily have u/(0) > 0. Suppose for contradiction that w attains its first maximum at t0 > 0 and w(t) < w(t0) for some t > t0. Then there are two cases: (i) w has a first minimum at some t\ > to, and (ii) w'(t) < 0 for all t > to and w(t) —> /3 £ [0, a) as t —> oo. We can easily derive a contradiction for the first case by using (7.11) and the fact w(t0) > w(ti), w'(t0) = w'(ti) = 0, since F(w(t0)) > F(w(ti)). The second case can be proved similarly, by

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127

replacing w(ti) with w(t) in the above argument and letting t —• oo. Hence w'(t) > 0 for t e [0,oo) and limt-,00 u>(t) = K £ (0,a]. We now show that K = a. Otherwise, 0 < K < a and there exists T 0 > 0 such that for t >T0, 0<

K/2

< w(t) <

K.

It follows that -10" = / ( « ; ) >

min

f(s) > 0 in [T 0 ,oo).

(7.12)

SS[K/2,K]

Clearly (7.12) implies w(t) —» oo as £ —* oo. This contradiction shows that K = a and w(t) —> a as £ —> oo. From this and (7.11), we deduce F(a) = (l/2)|w'(0)| 2 , and hence i|W'(i)|2+JF(u,(t))=F(a). Since w'(t) > 0, the above identity implies that w(t) is uniquely determined by ' y/2t for all t > 0 if (F5) holds,

f

dw

J0

[F(a) - F{w)]V*

w{t)

y/2t for t 6 (0,t 0 ), and w(t) = a for t > t0 if (F 4 ) holds,

^'l

'

where

"-u^tm^k

F{w)]1/2'

The proof is complete.

D

Proof of Theorem 7.1. Let u be a bounded positive solution of (7.1). By Lemmas 7.6 and 7.7, we have u < u < u in T. On the other hand, the minimality of u and the maximality of u force them to be functions of the variable x\ only, as the equation is invariant under translations in (x2, ...,xn). Thus, both u and u are positive solutions of (7.10). By Lemma 7.9, we necessarily have u(x) = w(xi) ~ u(x) for x € T. This implies u(x) = w(xi)

for x £ T.

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The proof of Theorem 7.1 is complete.

•

Proof of Theorem 7.3. This is a direct consequence of Theorem 7.1 and the formula for w(t) in (7.13). D Remark 7.10 All the results in this section can be extended to the pLaplacian case; see [Du-Guo(2004a)j. Moreover, these symmetry results remain valid if the boundary condition U\QT = 0 in (7.1) is replaced by U\QT — a> where a is a positive constant or oo. See [Dong(2004a)] for details. Remark 7.11 It has been conjectured by Berestycki, Caffarelli and Nirenberg in [Berestycki-Caffarelli-Nirenberg(1997a)] that for any C 1 function f{u), every bounded positive solution of (7.1) is a function of X\ only. This conjecture has been shown to be true in dimension N — 2 in [Berestycki-Caffarelli-Nirenberg(1997a)]; for N = 3, this can be proved under one extra condition /(0) > 0 (see [Berestycki-CaffarelliNirenberg(1997a)]); for N = 4, this is proved in [Alberti-AmbrosioCabre(2001)] under the extra condition that / is nonnegative in [0, oo) or there exists a > 0 such that / > 0 in [0,a] and / < 0 in [a, oo). The general case remains open.

7.2

Uniqueness results of logistic type equations over RN

Consider the logistic type elliptic equation -Au = a{x)u - b(x)uq,

x G RN,

(7.14)

where q is a constant greater than 1, a(x) and b(x) are continuous functions with b{x) positive on RN. By a positive solution to (7.14), we mean a function u € C1(i?-'v) satisfying /

Du-D4>dx= N

JR

[a{x)u-b{x)uq]<j)dx

f

Vcj> £ C%°(RN).

N

JR

Equations of this kind have attracted extensive research because of interests in mathematical biology and Riemannian geometry (see [AfrouziBrwon(1998)], [Bianchini-Rigoli(1997)], [Dong(2004b)], [Du-Ma(2001)], [Du-Ma(2002)], [Ratto-Rigoli-Veron(1997)] and the references therein). We will consider the uniqueness problem here under the following assumptions on a(x) and b(x):

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129

There exist 7 > - 2 , r £ (—00,00), and positive numbers 0:1,0:2 a n d /?i, 02

sucn

that

<*i = H m | a . | _ 0 0 ^ , o 2 = lim| I |_ > 0 0 gl

-y J

(7.15)

We want t o prove t h e following result. T h e o r e m 7 . 1 2 Suppose that (7.15) holds. Then (7.14) has a unique positive solution. Moreover, the unique positive solution satisfies u9-1(x) oi l ^ H c o ^ p r > £ ,

—h m

,

ui~llx) ^ ^ ^

-

a2 .

.„„„, (7.16)

If a(x) = a and b(x) = 6 are positive constants, it follows from t h e above result t h a t the only positive solution of the problem ~Au

= au~buq,

xeRN

(7.17)

is t h e constant (a/b)1^q~1\ This m a y be regarded as a Liouville type result. It h a s been shown in [Du-Ma(2001)] t h a t (7.17) h a s no positive solution if a < 0. One might like to compare these results with t h a t in Theorem 5.1. Our strategy for t h e proof of Theorem 7.12 is t h e following: We first prove t h a t (7.14) has a positive solution a n d every positive solution satisfies (7.16), we then use (7.16) and a Safonov type iteration technique t o derive a contradiction if there are more t h a n one positive solutions. In order t o prove (7.16), we need t h e following two lemmas. L e m m a 7 . 1 3 Let Q be a bounded domain in M.N with smooth boundary andL be as in Lemma 5.6, namely, Lu = ^ij[ciij(x)uXi]Xj withaij £ L°°(£l) satisfying, for some C2 > c\ > 0, aij = aji,

CJC 2 < £yay-(a;)&£,- < c 2 |£| 2 Vrr £ ft, £ €

RN.

Suppose a and (3 are continuous positive functions on ft, q > 1 and let fix denote the first eigenvalue of —Lu = \ia(x)u onQ, under Dirichlet boundary conditions on dfl. Then the problem -Lu

= fj,u[a(x) - P(x)uq-X],

w|an = 0

(7.18)

has a unique positive solution for every fi > (ii, and the unique positive solution Ufj, satisfies u M (x) —* \a(x)/(i{x)]1^q~x^ uniformly on any compact subset of ft as fj, —> 00.

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Applications

Proof. The existence follows from an upper and lower solution argument. Clearly any constant greater than or equal to M = maxQ-[o;(x)/(3(x)]1^p~1^ is an upper solution. Let <j> be a positive eigenfunction corresponding to Hi; then for each fixed /x > n\ and all small positive e, e < M and is a lower solution. Thus there is at least one positive solution. If u\ and «2 are two positive solutions, we apply Lemma 5.6 and conclude that U\ < Ui and U2 < Mi both hold on f2. Hence 1x1 = «2- This proves the uniqueness. For the asymptotic behavior with large /x, we first prove it under the extra condition that a(x) and f3(x) are smooth. Given any compact subset K of fl and any small e > 0 such that e < v0 = [a(x)//3(x)]1^q~1^ on Q, we let ve = vo + e, and find that ve(a(x) — f3(x)vq~1) < —6 on Q. for some positive constant 5 = 5(e) and — Lve > — c on fi for some positive constant c = c(e). It follows that for all large ji, ve is an upper solution of our problem. On the other hand, let <j> be the positive eigenfunction corresponding to /Ui with ||(/>||oo = 1. Then we can find a small neighborhood of dfl in fi, say U, such that is very small in U so that for all \i > ^ i + 1, —L<j> — fj,ia(x) < n4>(a(x) - /3(x)(j)q^1) on U. By shrinking U further if necessary, we can assume that U D K = 0 and < VQ — e on U. Now we can choose a smooth function we on Q such that we = on U, we = VQ — e on K and i>o — e/2 > we > 0 on the rest of CI. It is easily seen that such we is a lower solution of our problem for all large fi. Since we < ve, we deduce we < Ufj, < ve on ft for all large /x. In particular, [a(a;)//3(x)]1/(
e

>

U/1

> [ a ( a; )//3(a;)] 1 /(9- 1 ) - e

on if for all large /x. This is to say that uM —> (a//?) 1 /^ - 1 ) as /x —> oo uniformly on if, as required. We now consider the general case where a and (3 are only assumed continuous. Then for any given small e > 0 we can find smooth positive functions ae and (3e such that \\a£ - a\\oo < e and ||/3e - /3||oo < e- For large /x, let ixj, denote the unique positive solution of (7.18) with (a,j3) replaced by (ae — e,Pe + e), and let u2^ denote the unique positive solution of (7.18) with (a,/3) replaced by (a£ + e,(3e - e). Applying Lemma 5.6 we see that for all large ji,

ul
u\.

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131

By what has been proved for the smooth coefficient case, we have

R - [ ( « , ( * ) - £ ) / ( & ( * ) + e)]1/(*-i), \ul^[(ae(x)+e)/(3e(x)-e)}^-V as n —> oo uniformly over compact subsets of CI. Therefore, for any compact subset K of CI, l i m ^ o o ^ ) < [(at(x) + e)/(3e(x) - e)] 1 /^-!) uniformly on K,

l i m ^ M * ) > [(ae(x) - e)/(Be(x) +

e)}1'^

uniformly on K. Letting e —» 0 in the above inequalities we deduce [a{x)/P(x)]1/{q~1)

limM_>00u^(a;) = l i m ^ ^ u ^ x ) = uniformly on K. L e m m a 7.14 problem

• Let CI, L, q, a and 3 be as in Lemma 7.13. Then the —Lu = fiu[a(x) — 8{x)up~l],

u\aa = oo

has a unique positive solution for each fj, > 0, and the unique positive solution Ufj, satisfies u^ —> {a/(3)1' (p~l> uniformly on any compact subset of Cl as \x —» oo. Proof. The existence and uniqueness part of the lemma follows from Theorem 6.18. Moreover, if we use u™ to denote the unique positive solution of -Lu = nu{a - Bup~l),

U\QQ = n,

(7.19)

where /x > 0 and n is a positive integer, then we know that u™ —> u^, as n —> oo locally uniformly in CI. As in the previous lemma, we first prove the asymptotic behavior of uM under the extra condition that a and 3 are smooth. Let K be an arbitrary compact subset of CI, VQ = (a/(3)1^q~1^ and e > 0 any small positive number satisfying e < vo on CI. It is easily seen that, for all large li, we = vo — e is a lower solution for (7.19) with n > wt. On the other hand, fix a no > 0 then we can find a small neighborhood U of dCl in CI such that uo = u^Q > VQ + e on U. Therefore, -Lu0

= fiou0(a - /3ul~ ) > fiuo(a - /3UQ~ )

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Applications

on U for all \i > JIQ. NOW let us choose a smooth function v€ satisfying ve = «o on U, ve = i>o 4- e on K and ve > vo + e/2 on the rest of Q. Then it is easily checked that v£ is an upper solution for (7.19) provided that /J, is large enough. Applying Lemma 5.6 we obtain we < un < ve on Q for all large fj, and every large n. It follows that w£ < u^ < ve on fi. This implies that Up —> vo on K as \x —> oo, as required. If a and (3 are only continuous, then we define ae and /3e as in the proof of Lemma 7.13, and define u1, v? analogously. By making use of (7.19) and Lemma 5.6, we can easily see that u* < u^ < u'\. The proof then follows the same argument as in Lemma 7.13. • We are now ready to prove (7.16). Lemma 7.15 Suppose that (7.15) holds. Then (7.16) holds for any positive solution of (7.14). Proof. By (7.15), for any given small e € (0,1/2), there exists RQ — Ro(e) > 0 such that a(x) > (a x - e)\x\\

b(x) < (/32 + e)\x\T V|x| > R0/2.

Let A : = {x : 1 - e < \x\ < 1 + e}, AR:

= {XGRN:(1-

e)R < \x\ < (1 +

e)R}.

Then for R> R0, we have a(x) > (ai - e)|i|T > aeR~<, b{x) < (/32 + e ) ! ^ < (3€RT Vx € AR, where a£ := (a x - e) min{(l - e)\ (1 + e)^}, 0e := {(32 + e) max{(l - e) r , (1 + e)T}. Therefore any positive solution of (7.14) satisfies, for R>

R0,

-&u > aeR^u - /3eRTuq in AR. We now consider the auxiliary problem -Az

= R?+2z{ae

- /? £ z« _1 ) in A, z\dA = 0.

(7.20)

By Lemma 7.13, (7.20) has a unique positive solution zR for all large R and zR —> ae/pe as R —> oo. Due to uniqueness, zR is radially symmetric.

Symmetry

and Liouville Type Results over Half and Entire Spaces

Define vR(r) = R^-T^^-lhR(r/R). -AvR

= aeRyvR

133

Then it is easily checked that

~ (3eRTvQR in AR, VR\9AR

- 0.

We now apply Lemma 5.6 and find that u > VR in AR whenever R > RQ. Therefore u(x) > vR(\x\) VX G AR, R > R0. Taking R = |a;|, we obtain

«w- £ ^Mri^ M ( i r , v w > i ! o , \xp-T

\xp-T

~

Therefore u(x)i-1 x

—\ \->°°

\x\-y-T

-1*1-00

'

,K

!

'

Letting e —> 0, we deduce

The remaining inequality in (7.16) can be proved similarly (by making use of Lemma 7.14) and is left to the reader. • Lemma 7.16 Suppose that (7.15) holds. Then (7.14) has at lease one positive solution. Proof.

For any R > 0 we consider the problem —Au = a(x)u — b(x)uq in BR(0),

w|asR(o) = oo.

By Theorem 6.18, this problem has a unique positive solution UR. Applying Lemma 5.6 we find that UR < UR> in BR>(0) if R > R' > 0. Therefore u(x) := lim/j—oo UR(X) is well defined and is a nonnegative solution of (7.14). We will show that u is not identically 0, and hence by the strong maximum principle, it must be a positive solution. Fix m > 1 and define A := {x G RN : 1 < \x\ < m}, AR := {x G RN : R < \x\ < mR}. Let 4> > 0 be the eigenfunction to —Au = Ai(^4)u in A, U\QA = 0-

134

Maximum

Then for each R > 0,

4>R(X)

-A0R

Principles

and

Applications

:= (x/R) satisfies

= \l{A)R-2cj)R

in AR, ^R\BAR

= 0.

Since 7 > —2, by (7.15) we find that, for all large R, a(x) > (a1/2)\x\~< > 2X1(A)R~2 Vz € AR. Therefore, for fixed large Ro and all small e > 0, < a(x)(e(j)R0) - b{x){e4>Ro)q in ARo, eRo\dARo = 0.

-A(ecpR0)

Applying Lemma 5.6 we deduce UR > ecf)R0 in AR^ for all R > ra.RoTherefore u > e<j)Ra in AR^ . This shows that u ^ 0 and hence is a positive solution of (7.14). • The following technical lemma is the core of our iteration argument to be used in the uniqueness proof. Lemma 7.17 Suppose that (7.15) holds and u\, u 2 are positive solutions of (7.14)- Then there exists R > 1 large so that, if x* E RN satisfies, for some k* > k > 1, \x*\>R,

u2(x*) > k*ui(x*),

then we can find y* £ RN, and positive constants Co = Co(R,k) and ro = ro(R,k) independent of x* andk*, such that \y* -x*\ = r o | a : * r 7 / 2 , u2(y*) > (1 + c0)k^u1(y„). Proof.

By (7.15) and Lemma 7.15, for all large R > 1 and \x\ > R,

(l/2)a1\x\''

,

(l/2)^\x\T

< b(x) < 2fo\x\T,

(7.21)

and, for i = 1, 2, H^t-rVto-l) < u.(x)

< A l 2 | a : |(7-r)/(,-l) j

( 7 2 2 )

where

m=d/2)y

,/^=2(^)

.

We now fix R > 1 large enough so that i ? " 1 " ^ / 2 ) < 1/2 and (7.21), (7.22) hold for all x satisfying \x\ > R/2. Then we define Q0 := {x e RN : u2(a;) > k*v,i(x)} n Br(z*)>

Symmetry

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135

where r = r 0 |a;»| _ T/ 2 , Br{x*) = {x € RN : \x - x*\ < r}, and ro € (0,1) is to be determined below. Clearly x £ fio implies \x*\ - r < \x\ < ]x*\ +r, which in turn implies, due to |:r*| > R and our choice of R, ( l / 2 ) | i . | < \x\ < (3/2)|i,|.

(7.23)

We now consider U2 —fc*«iin Q0. Using (7.21), (7.22) and (7.23) and the assumption that u 2 > k*ux in f2o> we deduce, for x e fio> A(M 2 — kfUi) = —a{x)(u2 — k+ui) + b{x){u\ — k*u\) > -a(x)(u2

- Kui) + b(x){klu\ - k*u\)

> -2a2\x\">{u2 - fc.ui) + {l/2)(3i\x\Tu\{kl > ~2a2\x^(u2

- Km) + ( l / 2 ) / ? i M ? k r

7

- fc„) +9(7_T)/(9_1)

( ^ - *.)

a

> —M|a;*| (u2 - k*ui) + mk„\xt,\ , where M = 2a 2 m a x { ( l / 2 ) \ (3/2) 7 }, a = r + 5(7 - r)/(q - 1), m = (l/2) ) 8 1/ if(fc«- 1 - l)min{(l/2) CT ,(3/2) CT }. With these preparations, we now define w(x) = (2N)-1mK\xt\"(r2

- \x - z*| 2 ).

Clearly w(x) > 0 in Br(x*) and Aw = — mk*\x*\a. x E Uo, A(u 2 -fc*wi+w) > - M i x * ! 1 \ u 2 - K u 1 ) > -M\x*y

It follows that, for

(^-Kux+w).

(7.24)

If we denote by Ai(fi) the first eigenvalue of —A over Q under homogeneous Dirichlet boundary conditions, we have Ai(fto) > \i(Br(x*))

= r" 2 Ai(Bi(a;»)).

Therefore Ai(fio) > r^ 2 |a;*| 7 Ai,

Maximum

136

Principles

and

Applications

where Ai = Ai(i?i(a;»)) is independent of a;*. We now choose ro € (0,1) small enough so that rQ2Xi > M and hence Ai(fi0) > M|x»| 7 . Then by the maximum principle in [Berestycki-Nirenberg-Varadhan(1994)] (see Theorem 2.8 if the underlying domain has Lipschitz boundary), due to (7.24), U2{x*) — k*ui(x*) + w(x*) < max(«2 — k^u\ + w). We observe that the maximum of (u2 — fc*ui + w) over dQo has to be achieved by some y* G dBr{x„) since any y G dQo \ dBr(x*) satisfies, by the definition of QQ, U2{y) = k*ui(y) and hence U2(y) - Kux(y) + w(y) = w(y) < w(x*) < u2(x*) - k*ui(x») + w(x*). Thus we can find y* G dflo satisfying |y» — x*\ = r (hence w(y*) = 0) such that U2(y*) - fe*wi(y*) = u2(y*) - k*ui(yt) + w(y*) > u2(xt)

- A;»ui(x») + w(x*)

> w(x*) =

(2N)~1mka,\x^\ar2

= (2N)-1mk,rl\x^~T

JA'" 1 )

where Cl

= (2N)-1mr2lmm{{l/2)-(-<-T)/('1-1\(3/2)-^-TV(q-V}

> 0,

and we have used (7.23). Making use of (7.22), we finally deduce u2{y*) - fc*«i(y*) > C I / C * | J / * | ( 7 _ T ) / ( 9 _ 1 ) > Therefore we can take CQ = cyp^

cm^ktUity*).

and the proof if complete.

We are now ready to prove Theorem 7.12 by a Safonov type iteration technique. Proof of Theorem 7.12. We have already proved in the previous lemmas that equation (7.14) has at least one positive solution and any such solution satisfies (7.16). Suppose by way of contradiction that (7.14) has

•

Symmetry

and Liouville Type Results over Half and Entire Spaces

137

two different positive solutions u\ and u2. Let — ui(x) —u2(x) fei = h m1^ i ^ ^ — - - , k2 = limui^oo—p—. '

ui(x)

«2W

By (7.16) we know that both fci and /c2 are finite. If k\ < 1 and &2 < 1, then for any e > 0 there exists Re > 0 such that for all x satisfying \x\ > R€, u\{x) < (1 + e)u2(x), u2(x) < (1 + e)ui(x). Since (l + e)ui and (l + e)u2 are upper solutions of (7.14), we apply Lemma 5.6 over Q = BR(0), R> Re, and deduce Mx)

< (1 + *)MX),

u2(x) < (1 + e)ui(x), Vx € i ? N .

Letting e —> 0 we obtain «i = u2, contradicting our assumption that they are different solutions. So necessarily max{fci,fc2} > 1- Without loss of generality we may assume that k2 > 1. Therefore there exist a constant k £ ( l , ^ ) a n d a sequence {x n } such that \xn\ -> oo, u2(xn)/ui(xn)

> k, n — 1, 2,...

We are now in a position to apply Lemma 7.17. Let R, ro and CQ be determined by Lemma 7.17. We recall that R satisfies i ? - * - ^ / 2 ) < 1/2. We first find an integer j > 1 such that ( l + c o ) ' ' * > sup ^ 4 4 . Since |x„| —> 00, we can then find no large enough such that

\xno\(l/2y > R. Taking x* = x n o and fc* = k in Lemma 7.17, we can find y* = y\ such that IS/x -a:*I = n ) | x » | " 7 / 2 , u 2 (j/i) > (l + c 0 )/cui(i/i). Clearly |2/i| > |x,| - r 0 | x , | - ^ 2 > |x n o |(l -

fi-1-^))

> |x n o |(l/2) > R.

We now take x* = yi and k* — (1 + Co)fc in Lemma 7.17, and we can find y2 such that 11/2 - yi| = »"o|yir 7/2 , u2(y2) > (1 + c 0 ) 2 fcui(y 2 )-

Maximum

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and

Applications

Let us note t h a t \y2\>\yi\(l/2)>\xno\(l/2)2>R. We can repeat the above process until we obtain yj, which satisfies u2(yj)

> {l + c0ykui(yj),

IJ/JI > | x n o | ( l / 2 ) J ' > R.

Therefore u 2(yi) ~ ^

>

,, si, U2(x) 1 + co) J k > sup — ) - ( .

This contradiction completes our proof.

•

R e m a r k 7.18 Our arguments in the above uniqueness proof can be extended t o cases where t h e right h a n d side of (7.14) is more general. For example, suppose t h a t , for u > 0, f(u)/u is increasing and c\ < f(u)/uq < c2 for some positive constants c\ and c2, and suppose furthermore t h a t l i m u ^ o f{u)/uq exists when 7 < r , liniu^oo f(u)/u<J exists when 7 > r , and c\ = c2 when 7 = r . Then the uniqueness conclusion in Theorem 7.12 still holds if uq is replaced by f{u) in (7.14). We leave the proof of this fact to the interested reader. R e m a r k 7.19 Once (7.16) is proved, one can also a d a p t the convex function trick used in the proof of Theorem 6.18, instead of the Safonov iteration technique used above, to prove the uniqueness of positive solutions of (7.14). This m e t h o d is much more simpler, but it seems difficult to apply to the more general cases mentioned in Remark 7.18 above. R e m a r k 7.20 T h e assumption t h a t 7 > - 2 in (7.15) played an import a n t role in our proofs. If a(x) < C|:r| 7 with 7 < —2 for some C > 0 and all large \x\, and a(x) is positive somewhere in RN, and b{x) > 0 in RN, t h e n the results in [Du-Ma(2002)] apply and by Theorem 1 there, there exists a unique G\ > 0 such t h a t -Au

= Xa(x)u - b(x)uq,

x € RN

has a unique positive solution u £ H when A > <Ti, and there is no such solution when 0 < A < <7i, where H denotes the completion of CQ(RN) 2 under the norm (fRN \Wu\ dx) . It is unclear whether there are positive solutions outside H, but for the special case b(x) s 1 and q = 2, it was shown in [Afrouzi-Brwon(1998)] t h a t , indeed, there are no other positive solutions.

Symmetry

and Liouville Type Results over Half and Entire Spaces

Remark 7.21 Lemma 7.15 can be extended to was first proved in [Dong(2004b)]. Lemma 7.14 Ma(2001)]; it was extended to the p-Laplacian However, it is unknown whether Theorem 7.12 p-Laplacian case.

139

the p-Laplacian case and was first proved in [Ducase in [Du-Guo(2003)]. can be extended to the

Remark 7.22 The Safonov iteration technique used in this section is taken from [Du-Liu(2005)]. Other versions of this technique can be found in [Kim(2002)], [Chuaqui et aZ.(2004)], [Du(2004b)] and [Cirstea-Du(2005)]. 7.3

Partial s y m m e t r y in the entire space

Consider the equation -Au = u-u3,

x£RN.

(7.25)

In [De Giorgi(1978)], De Giorgi stated the following conjecture: If N > 2 and u is a solution of (7.25) such that \u{x)\ < 1, uXl(x) > 0 in RN, then u is a function of one variable only; that is, there exists a unit vector v € R and a one variable function 4> such that u(x) = 4>(v • x). It is easy to show that the solutions of -0" = 0-03,

tGR1

are 4>(t) = tanh(t/\/2 + p), /i £ i? 1 is an arbitrary constant. Therefore, if this conjecture is true, then all the solutions of (7.25) are given by u ( z ) = t a n h [ ( l / V 2 > - a : - r / i ] , t / S S ^ " 1 , /x G R1. This kind of results is useful in understanding sharp transition layers in a variety of problems. Let us observe that the restriction |u(:r)| < 1 is unnecessary. Indeed, we have the following result. Lemma 7.23 sense that f

N

Suppose that u £ C1^1*)

Du • D(f>dx = f

JR

JR

N

N

Then \u(x)\ < 1 in R .

(u-

is a solution of (7.25) in the

u3)<j)dx V<£ e C%°(RN).

Maximum

140

Proof.

Principles

and

Applications

Given any R > 0 and XQ € RN, we consider the problem -Av

= v - v3 in BR(x0),

V\9BR(XO)

= oo.

(7.26)

By Theorem 6.18, (7.26) has a unique positive solution VR. We claim that any solution u of (7.25) satisfies u{x) < VR(X) in BR(XO). Otherwise, the set D := {x £ BR(XO)

: u(x) > VR(X)}

is nonempty and on any component

Do of D, we have u{x) > VR{X). On the other hand, since u — VR on ODQ, we can apply Lemma 5.6 to conclude that u < VR in Do- This contradiction proves our claim. Therefore U(XQ) < VR{XQ). Define WR(X) = VR(XQ, 4- Rx) for x £ Bi (0); we have -AwR

= R2wR(l

- w2R) in -Bi(O), wR\aBl{0)

= oo.

By Lemma 7.14 we find that WR(X) —> 1 as R —> oo locally uniformly in Bi(0). In particular, WR(0) —» 1 as i? —> oo. It follows that UR(:TO) —> l as R —> oo and hence, letting i? —> oo in u(xo) < VR{XQ) we deduce M(XO) < 1. Since XQ is arbitrary, we thus have proved that u{x) < 1 in i?-^. Since — u satisfies (7.25), we can apply the above argument to — u and obtain — u < 1, that is, u > —1. • The conjecture was proved by Ghoussoub and Gui [GhoussoubGui(1998)] for the case N = 2. Actually they proved the following more general result. T h e o r e m 7.24 of

If f e C 1 ^ 1 ) and u e L°°(R2) n C2(R2) is a solution

-AM

= f(u),

x e R2,

such that uXl > 0 in R2, then u is a function of one variable only, that is, there exist constants a, b and a one variable function 4>{t) such that 1 U{XI,XQ) = (f>(axi + bx2). Clearly <j> must satisfy —" = f((f>) in R . We will prove this result by arguments given in [Berestycki-CaffarelliNirenberg(1997a)]. A key step is the following Liouville type theorem. T h e o r e m 7.25 Let 4> £ Lf£c(RN) be a positive function. o~ £ WtgC (RN) satisfies in the weak sense o-div(2Da) > 0 in RN,

Assume that (7.26)

and for some C > 0 and every R > 1, f JBR{0)

{^afdx

< CR2.

(7.27)

Symmetry

and Liouville Type Results over Half and Entire Spaces

141

Then a is a constant. Proof.

From (7.26) we deduce, for any smooth function i/>, div(ai)2(j)2Da) > ip24>2\Da\2 + 2o-ip2D4> • Da.

(7.28)

Let C be a C°° function on [0, oo) with 0 < ((t) < 1 and £(£) = 1 for 0 < t < 1, C(t) = 0 for t > 2. For R > 0 and x £ RN, set (R(x) = ((\x\/R). Taking ip = (R in (7.28) and integrating over RN, we find, by the divergent theorem, /

CU2\Da\2dx<2

f

JRN

cfCRaDCR-Do-dx

JRN

<2

JR. >R<\x\<2R

1/2

A2^2 (R\Da\'dx

/

JRN

0V|Z?c*|5 dx

1/2

By (7.27) and the definition of (R, we can find C\ > 0 such that

JRN

Therefore

I

CR4>2\Da\2dx < 2sJCx

f

k2s2 Q\Da\2dx

1/2

(7.29)

R<\x\<2R JR<\X

This implies that CRP\Da\2dx

JRN

< Ad,

and hence, letting R —> oo in (7.29) we obtain

I

b2\Da\2dx = 0.

RN

This implies |Dcr| = 0 and hence a is a constant.

•

Proof of Theorem 7.24- Let u be a bounded solution of —AM = f(u) in R2. Then f{u{x)) is bounded over R2. Applying interior elliptic estimates over an arbitrary unit ball in R? for u, we deduce that |£)M(:E)| has a bound independent of a; € R2, say \Du(x)\ < M for all x £ R2. Now let <j>{x) := uxi(x). By assumption, > 0 and by differentiating the equation for u with respect to x\ we find -A = /'(«)
142

Maximum

Principles

and

Applications

Therefore, we can apply the strong maximum principle to deduce that either = 0 on R2 or 4>{x) > 0 everywhere. In the former case u(x) is independent of x\ and hence is a one variable function. In the latter case, 2Da) =

^A(MT2)

- uX2A(f> = f'(u)(uXluX2

- uXluX2)

- 0.

Moreover, /

{4>afdx=

JBR(0)

(uX2)2dx<M2\BR(0)\
f JBR{0)

Hence we can apply Theorem 7.25 to conclude that a is a constant, that is, uX2 = cuXl for some constant c. This implies that dvu = 0 for v — (—c, 1). Therefore u is a function of xy + cx2 alone. D The above arguments have been refined by Ambrosio and Cabre to show that Theorem 7.24 holds for N = 3; see [Ambrosio-Cabre(2000)] for details. Coming back to equation (7.25), it was shown by O. Savin [Savin(2003)] with a very different method that any bounded solution satisfying uXl > 0 must be a function of one variable provided that N < 8 and for any fixed x' := (x2,

...,XM),

lim

u(xi,x')

= ± 1 . (7.30)

Xl—»±00

More general results can be found in [Valdinoci-Sciuzi-Savin(2005)]. The restriction N < 8 can be removed if the other condition in (7.30) is strengthened to lim

u(xi,x')

= ±1 uniformly for x'.

(7.31)

Xl—>±QO

In fact, under (7.31), the condition uXl > 0 can also be removed. The following theorem is due to Berestycki, Hamel and Monneau [BerestyckiHamel-Monneau(2000)]. Theorem 7.26 Suppose that u is a solution to (7.25) and satisfies (7,31). Then u is a function of x\ only. The proof of Theorem 7.26 relies on the following version of the maximum principle in unbounded domains; see [Berestycki-CaffarelliNirenberg(1997b)]. Lemma 7.27 Let D be a domain in RN, possibly unbounded. Assume that D is disjoint from the closure of an infinite open connected cone S.

Symmetry

and Liouville Type Results over Half and Entire Spaces

143

Suppose that u G C(D) D Wio'c (D) is bounded from above and satisfies in the weak sense, for some nonnegative function c(x), —Au + c(x)u < 0 in D,

U\QD

< 0.

Then u < 0 in D. Proof. We may suppose that the vertex of £ is the origin. By narrowing S we may suppose that D lies outside a slightly larger cone, and dT, is smooth away from the origin. Let ip > 0 be the principal eigenfunction of the Laplace-Beltrami operator —As on the unit sphere S = S' iV_1 in the region G := SN^ \T,: - A S V = MiV> in G,

I/J\9G

= 0.

(Similar to the theory for —A over a bounded domain in RN, the principal eigenvalue /J.I of the above problem is positive, which is the only eigenvalue corresponding to a positive eigenfunction.) We now choose a > 0 such that a(JV + o ; - 2 ) = m. Clearly any x G RN can be uniquely written as x = r£ with r > 0 and £ G 5. Thus, if x = r£ G S c := RN \ S, then £ G G. Consider in E c the function jW = r ° ^ U e G , r > 0 . It is positive in D and - A g + eg > -Ag = ra-2[a(n

+ a - 2)i/> + Asip] = 0.

Let a = u/g. Then in D,

Ao-+*Dg.Do-+{A-c)9*>0. 9 9 On the boundary dD, a < 0, while lim|x|_>ooxe£)(7(a;) < 0. Since (A — c)g < 0 in D, we can argue as in the standard maximum principle to conclude that a < 0 in D. Hence u < 0 in D. • Remark 7.28 The condition that u is bounded from above can be relaxed. From the proof, we see that it is enough to assume that lim^i^oo tX £D u { x )I'\ x \ a ^ 0i where a is determined by £ as in the above proof. Moreover, when N = 2, the condition on D can be relaxed. It is enough to assume that D ^ R2. For such a case, we can replace g in the above proof by C + In \x — XQ\, where XQ $ D and C is a large positive constant.

Maximum

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Applications

Proof of Theorem 7.26. Denote f(u) = u — u3. We can find S > 0 small such that f'(u) < 0 in [1 - 6,1 + 6} U [-1 - 6, - 1 + <5]. Let u be a solution of (7.25) satisfying (7.31). By Lemma 7.23, we know that \u\ < 1 in RN. By (7.31) we see that u ^ ± 1 . Therefore we can apply the strong maximum principle to conclude that — 1 0. To this end, we fix such a v and for t € R1 define u*(x) = u(x + tv). Clearly - A u * = /(u*) in # N . By (7.31), there exists a > 0 such that u{x) G [1 — 5,1) when Xi > a and u(x) € (—1, —1 + 5] when xi < —a. It follows that, for t > 2a/ux, uf{x) € [1 — S, 1) for x\ > —a. We now consider v := u* — u with t > 2a/v\. From the above discussion, we have W(T) > 0 when x\ = —a. If {x £ RN : v(x) < 0} is nonempty, then any component D of this set satisfies D C {x e _RW : xi < —a} or Z) C {x € i?^ : xi < —a}. In the former case, we have — 1 < w'(x) 0 in D, v\8D = 0.

By Lemma 7.27, we deduce v > 0 in D, which is a contradiction. Therefore M* > u in i?^. We claim that M' > M in RN for all t > 0. To prove this, we define r = inf{s > 0 : uf > u in RN for all t > s}. We already know that T > 2a/v\. Suppose r > 0, we are going to derive a contradiction. By continuity, we have uT > u in RN. If [uT(x) - u(x)} = a0 > 0,

inf

(7.32)

—a<xi
then for 770 > 0 small enough and t £ [T,T — r/o], u*(x) — u(x) > w r (x) — M(X) + sup IDwI^j -1 ^ — T\ RN 1

> O-Q- rjoi/i sup \Du\ > 0, Vxi e [-a, a]. Since u(x) > 1 — S for xi > a, it follows that M*(X) > 1 — S when xi > a and i > 0. We can now repeat the above argument for v to deduce that M' > u in i? w for all t £ [T, T — 770]. But this contradicts the definition of r. Thus (7.32) is impossible and we necessarily have inf

[uT(x) - u{x)} = 0.

(7.33)

Symmetry

and Liouville Type Results over Half and Entire Spaces

145

T h e n there exists a sequence {xn} C {x £ R : — a < x\ < a} such t h a t T n n n u (x )-u(x ) —> 0 as n —> oo. Set un(x) = u(x +x). Then -Aun = f(un) in RN. Since f(un) is bounded in RN, from s t a n d a r d elliptic estimates we know t h a t {«„} is a bounded equi-continuous sequence in RN. Hence by passing to a subsequence, we may assume t h a t un —> u^ locally uniformly in RN. Moreover, u^ satisfies (7.25). It follows t h a t uTn —> u^, u^ > ux and u^o(0) = Uoo(0)- T h e function v := u^ — u^ satisfies -Av

+ c(x)v = 0, v > 0 in RN,

v(0) = 0,

where c(x) = [/(u^x)) - /(uoo(a;))]/[u^(a;) - Uoo(z)] if w ^ ^ ) 7^ "00(2;), and c(x) = 0 otherwise. Since / is C 1 and ul0(x),u00(x) G [—1,1], we find N t h a t c(x) is bounded in R . Therefore we can apply the strong m a x i m u m principle to conclude t h a t v = 0 in RN. This means t h a t Moo(^) = uoc(x -f TV). Hence Uoo(x) is periodic with respect to the vector £ := rv. On the other hand, since {xn} C {2; G RN : —a < x\ < a}, it follows from u satisfying (7.31) t h a t Uoo satisfies (7.31). Since TV\ > 0, (7.31) implies t h a t UQQ cannot be periodic with respect to TV. This contradiction shows t h a t (7.33) cannot hold either. Therefore, we must have r = 0. This implies t h a t u{x) is increasing in any direction v with V\ > 0. From continuity, we find t h a t dvu > 0 for any v with v\ > 0, in particular, for any v* with v\ = 0. Replacing u* by — 1/*, we deduce du*u — 0 for any v* with first component 0. Therefore u is independent of Xi, i = 2, ...,N, and there exists 4>{t) such t h a t u(x) = 4>(xi). • R e m a r k 7.29 Theorem 7.26 holds for more general equations t h a n (7.25); see [Berestycki-Hamel-Monneau(2000)] for details. T h e general case of the De Giorgi conjecture is still open.

7.4

S o m e Liouville t y p e results

We discuss here several Liouville type results which are useful for obtaining a priori estimates for positive solutions of semilinear elliptic equations over bounded domains; some of these equations will be studied in detail in Volume 2. We first consider the problem - A M = up, u > 0 in RN, where p > 1.

(7.34)

Maximum Principles and Applications

146

Theorem 7.30 Suppose that N > 3. If 1 < p < (N + 2)/(JV - 2), then u = Q is the only C2 solution of (7.34). If P = iN + 2 ) / W - 2), then any positive C2 solution of (7.34) *s radially symmetric and hence assumes the form U{X)

= (x2

+

\x-x0\2)

'

A

> °^°

£

* •

Proof. Suppose that u £ C2(RN) is a positive solution of (7.34) with 1 < p < (-^ + 2)/(N — 2). We will use the moving plane method. Since no growth condition on the solution u is made at infinity, we use the so called "Kelvin transformation" to define a new function

«(*) = ! ^ P . ^ ^ \ { 0 } . It is easy to verify that v satisfies the equation -Av

= \x\avp, xeRN\

{0}, a = p(N - 2) - (N + 2) < 0.

(7.35)

The idea is to show that v is radially symmetric about some point and then so is u. We start by moving planes in the x\ direction. To do that we define, for Ae-R 1 , S A = {x e RN : xx < A}, Tx =
=

(2X-xi,x2,...,xN),

so that x* is the reflection point of x about the plane T\. We will use the notations v$x) = v(xx),

wx(x) = v\(x) - v(x), w\{x) =

w\(x)/g(x),

where g{x) = ln(3 — x$. Since v(x) is singular at the origin, u>\ is singular at the origin and at x° := (2A, 0, ...,0). In the following, we will consider w\(x) with A < 0. For such A, the only possible singular point of w\(x) in SA is x°x. So we will consider u>\ in Y,\ := T,\ \ {x°x}. Step 1. eo := inf 0 < | x | 0. Since -Av > 0 in Bi(0) \ {0} and v > 0 on dBx(0), for any r e (0, l ) , we can apply the maximum principle to conclude that v > vr in Ar := {x £

Symmetry

and Liouville Type Results over Half and Entire Spaces

147

RN : r < \x\ < 1}, where vr denotes the unique solution of -Avr

= 0 in Ar, vr\{\x\=i] - £, vr\[\x\=ry

= 0,

with £ > 0 small enough so that v|{|.,.|=1}. > £ . The maximum principle also shows that vr is decreasing in r and vr < V in Ar for all r £ (0,1), where V (= £) is the unique solution of - A V = 0inB1(0),

V\dBl{0)=C

Therefore VQ(X) := lim r ^o vr(x) is well-defined and VQ < V in -Bi(O) \ {0}. It follows that v > v0 in Bi(0) \ {0}. On the other hand, by uniqueness, we know that each vr and hence VQ is radially symmetric. Therefore VQ = vo(t) satisfies (rN~1v')' = 0 for t £ (0,1] and i>o(l) = £. A simple integration then yields that 1-

t2~N

for some constant c. Since 0 < vo < V, we necessarily have c = 0 and hence «o(t) = £• Step 2. For A < 0 negative enough, w\{x) > 0 for x £ £,\. Arguing indirectly, we assume that there exists A„ —> — oo such that inff w\n(x) < 0. From the definition of v we find that v(x) —> 0 as |a:| —* oo. Therefore, for all large n, v(x) < eo when x 6 B\{x^ ), where eo is given in Step 1. This implies that wXn{x) > 0 in B\{x°x ) \ { i ^ } for all large n. For any fix such large n we notice that wXn (x) —* 0 as \x\ —• oo, and U)A„ (Z) = 0 for x £ <9EA„ • Therefore inf j wXn (x) is achieved by some

x-etXn\B1(x0xJ.

Since A„ < 0, \xXn\ < |x| for x £ EA„ and hence -AvXn{x)

> \x\avXn{x)p

for i e S i „ .

It follows that AwXn + c(x)wXn < 0 in E A „,

(7.36)

and hence AwXn + -Dg • DwXn + \c(x) + - 2 ] «jA„ < 0 in E A „, (7.37) 9 9 where c(x) = p\x\aip(x)p~1 and tp(x) lies between v(x) and i>(:rA™). Since n z € SA„ \ -Bi(a;° ) and A„ —* - o o , we know that |a;"| —> oo as n —* oo,

Maximum

148

Principles

and

Applications

and ( x n ) A " g" B i ( 0 ) . It follows t h a t d l x " ! 2 - ^ < c i K a : " ) * " ! 2 ^ < vXn(xn)

< (*") < v{xn)

<

c2\xn\2-N

for some C2,ci > 0. This implies t h a t c(x") < cala;"! - 4 for some C3 > 0, and hence c(l") + A9(x")/9(x") < c3|*T< -

( 3

,

i ; ) 2

;

n ( 3

_

1 ? )

< 0

for all large n. Since wXn has a negative minimum a t xn, AwXn(xn) > 0 and Dw\n(xn) = 0. Therefore, if we take x = xn in (7.37), the left h a n d side of the inequality is positive for all large n. This contradiction proves our conclusion in Step 2. Step 3. Let A0 :— sup{/x < 0 : w\ > 0 in £ A for all A < fi}. T h e n wXo = 0 in E A o if Ao < 0. By Step 2 we know t h a t Ao is well-defined. By continuity, wXo > 0 in £ A o . Suppose t h a t wXo ^ 0 in E A o . wXo also satisfies (7.36). Applying the strong maximum principle to (7.36) we deduce wXo(x)

> 0 in £ A o and dXlwXo

< 0 on TXo.

(7.38)

(We note t h a t x°Xo £ TXo due to A0 < 0.) By the definition of Ao, there exists a sequence A^ decreasing to Ao such t h a t w\k{x) < 0 for some x G ^xk- We now consider infjwXk(x). If 5 > 0 is chosen small enough, we have Bsix^) C £ A o and hence wXa > 0 in Bs{x°Xa) \ {x°Xo}. It then follows from (7.36) t h a t - A w A o > 0 in Bsix^ ) \ {x° }. T h e argument in the proof of Step 1 then shows t h a t w\0 and hence w\0 is bounded from below over B${x0x ) \ {x°x } by some positive constant e. This implies t h a t for all large k, w\k (x) > e for x € Bj/2{x\k). Since w\k(x) —» 0 as |x| —> 00 for any fixed k, and wXk{x) — 0 for x G T\k, we know t h a t for each large k, inif. wXk(x) is achieved by some xk G T,Xk \ 5 j / 2 ( x ° f e ) . Moreover, using (7.37) and the estimate for c + Ag/g in Step 2, we know t h a t there exists some Ro > 0 such t h a t \xk\ < RQ. Therefore, by passing to a subsequence, we m a y assume t h a t xk —> x° as k —> 00. It then follows from wXk{xk) < 0 and DwXk(xk) —0 that wXo(x°)<0,

DwXo{x°)

= 0.

(7.39)

From xk G £ Afc \ Bs/2(x°Xk) we deduce x° € £ A o \ -B 5 / 2 (^° )• B u t then (7.39) is in contradiction with (7.38). This completes our proof for Step 3.

Symmetry

and Liouville Type Results over Half and Entire Spaces

149

Step 4- Completion of the proof. From Step 3, we know that when we increase A towards 0, we either meet some A0 < 0 such that u>\0(x) = 0 in £>,0, a n d hence v(x) is symmetric about xi = Ao, or we have u>\(x) > 0 in EA for all A < 0 and hence wo{x) > 0 in So- In the latter case, we can carry out the above procedure in the opposite direction, namely moving the parallel planes in the negative x\ direction from positive infinity towards 0. If our planes T\ stop somewhere before the origin, we obtain symmetry of v(x) about some X\ = Ao > 0. Otherwise we obtain an inequality which is equivalent to WQ{X) < 0 in SO and hence, combined with our earlier inequality we deduce wo(x) = 0 in SoThis implies that v(x) is symmetric about x\ = 0. Therefore in either case, v(x) is symmetric about some plane perpendicular to the xi axis. Since (7.34) is invariant under rotations of the coordinates, we can choose the x\ axis as any given direction. Our previous conclusion then implies that v{x) is radially symmetric about some point XQ in RN. If p < (N + 2)/(N — 2), then a < 0 and from the equation satisfied by v we find that v(x) > 0 can only be radially symmetric about the origin. This implies that u(x) is radially symmetric about the origin. Since (7.34) is invariant under translations of the variable, we may choose any point as the origin, and hence our above conclusion implies that u is radially symmetric about any point in the space. This is possible only if u = c, a constant. Then from the equation we necessarily have c = 0. This contradicts out assumption that u is a positive solution of (7.34). Therefore there is no positive solution. By the maximum principle, the only nonnegative solution is the trivial solution u = 0. If p = (N + 2)/{N — 2), and if v is radially symmetric about the origin, then so is u(x). If v{x) is symmetric about some point other than the origin, then lim|x|_»0 v(x) i s well-defined and hence \x\N~2u(x) —> c > 0 as \x\ —> oo for some c > 0. With the existence of this limit, u behaves like v at infinity and we can apply to u the above moving plane argument used for v to deduce that u is radially symmetric about some point. Therefore u is always radially symmetric about some point in RN. Finally, the radial symmetry of u and the known uniqueness of the solutions of the corresponding ordinary differential equations (see [Cerverno-Jacobs-Nohl(1977)]) give the formula for all the positive solutions of (7.34) with p = (N + 2)/(7V - 2). • Theorem 7.30 was first proved by Caffarelli, Gidas and Spruck in [Caffarelli-Gidas-Spruck(1989] and [Gidas-Spruck(1981)]. Our proof above follows the one given by Chen and Li in [Chen-Li(1991)], which considerably

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simplified the original one in [Caffarelli-Gidas-Spruck(1989] and [GidasSpruck(1981)]. For dimensions N = 1,2, better results can be obtained. We consider the case N = 2 next, and it is more convenient to consider the case N = 1 when we discuss the half space problem later. Theorem 7.31 sense

Suppose that u g Wj0'c (BR)(~\C(BR)

satisfies in the weak

- A w > up, u > 0 in BR, where p>\

and BR := {x 6 R2 : )x\ > Rj, R> 0. Then u = 0.

Proof. Suppose that there exists some nonnegative nontrivial u S w loc(BR) n C{BR) which satisfies - A u > up in BR. By the strong maximum principle, we find u > 0 in BR. We divide our discussions below into three steps. Step 1. lim| THoo M(a;) > 0. For S > R we define, for x € BR, v(x] [X)

- W4g/N) ~ ln(2S/R) •

It is easy to see that —Av = 0 and v = 1 when \x\ = 2i?, v = 1/2 when |a;| = 2(2RS)1'2,

v = 0 when |x| = AR.

Since u > 0 in i? f l , we have e := inf|x]=2ijw > 0. Hence we can apply the weak maximum principle to conclude that u > ev for 2R < \x\ < 4 5 . It follows that u(x) > e/2 for 2R < \x\ < 2(2RS)1/2. Letting S -> oo we deduce u{x) > e/2 for |x| > 2R. This proves Step 1. Step 2. If B2r(x0) C BR, then /"

u p_1 da; < C

for some constant C independent of r and x$. Let £ be a radially symmetric C 2 cut-off function on the ball i?2(0), namely, (i) £ = 1 for |x| < 1; 0 < £ < 1 for |x| > 1, (ii) ^ has compact support in -B2(0), (iii) | ^ | < 2, |A^| < c.

Symmetry

and Liouville Type Results over Half and Entire Spaces

Denote £r{x) = £{\x — xo\/r) and take = £2u and we obtain gu-2\Du\2dx+

f

f

1

> 0 as a test function

gvP-Hx (7.40)

u-lDu-D{g)dx.

<[

151

JB'jr.txn)

Since

iD(ai=2^i^ P |<^ ! "sr

have

u-^Du- D(£) <

eri^lDuDf^-) \Tq,r J

<(l/2)eru-2\Du\2

+ Cr-2

for some C > 0. In turn, by (7.40), (1/2)/

gu-2\Du\2dx+

&v?-1dx<

f

JB-ir(xa)

JB2r(x0)

By our definition, £;? = 1 in Br(xo). implies f

Cr'2dx:=Cv

f J

B2T{X0)

Therefore, the above inequality

up~ldx < Cx.

(7.41)

JBr{x0)

This finishes the proof of Step 2. Step 3. Completion of the proof. Take a sequence {xn} C R2 such that \xn\ > 3-ff and \xn\ —> oo. By (7.41), / uP^dxKCu

n = l,2,...,

JB

where B = B\Xny±(xn). mlinu^Bi n i B

Then 1

< IBI- 1 f up~ldx < C2\xn\-2

-» 0

JB

as n —> oo, contradicting our conclusion in Step 1. Therefore there are no nontrivial nonnegative solutions. •

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Applications

Remark 7.32 Theorems 7.30 and 7.31 are special cases of more general results for p-Laplacian equations obtained recently by Serrin and Zou; see [Serrin-Zou(2002)] and the references therein for further related results. We now consider the half space problem - A u = up, u > 0 in R%, u\dRN = 0,

(7.42)

where p > 1 and R% :={x = (xi,...,xN)

eRN

: x i > 0}.

Theorem 7.33 Suppose that u(x) is a C2{RX) solution of (7.42) with Kp<(N + 2)/(N - 2) when N > 3, and with 1 < p < oo when N = 1,2. Then u(x) = 0. Proof. By the strong maximum principle, it suffices to show that (7.42) has no positive solution. Suppose by way of contradiction that it has a positive solution u. We are going to derive a contradiction. We start with the case N > 3 and first show that u{x) depends only on x\. We will write x = {xi,x') = (XI,X2,...,XN). For two points x,x G i?+ with x\ = xi, we will show that u(x) = u(x). To this end, we shift the origin to the point (0, (x'+x')/2) while keeping the plane xi = 0 unchanged. In the new coordinate system we perform an inversion about the point - e 1 = (-l,0,...)0), y={x

+ e1)/\x + e1\2, xt>0

(7.43)

and define

«,(») = \x + e i r V * ) = U{~e}tVl}V?).

(7.44)

Under this transformation, the half-space x\ > 0 becomes the ball \y — e i / 2 | < 1/2, the boundary x\ = 0 goes to the sphere \y — e\/2\ = 1/2, and the point x = oo goes to y = 0. The function w(y) satisfies, for a = p(N-2)-(N + 2), - A w + \y\awp(y), w > 0 in \y - exj2\ < 1/2

(7.45)

and, by (7.44), w(y) = 0 on \y-e\/2\ = 1/2 except possibly at y = 0 where w(y) may be singular. We are going to use the moving plane method to show that w(y) is axisymmetric about the yi-axis, i.e. w = w(yi, \y'\). It suffices to show that for any direction perpendicular to the yi-axis, w(y) is symmetric with respect to that direction. As usual, since problem (7.45)

Symmetry

and Liouville Type Results over Half and Entire Spaces

153

is invariant under rotations about the j/i-axis, it suffices to show this for one direction, say the y^ direction. Since p < (N + 2)/(N — 2), a < 0 and hence the coefficient \y\a is non-increasing in the y% direction (in fact in any direction orthogonal to the yi-axis) as y varies in the ball \y — e\j2\ < 1/2. Due to this monotonicity property, we find that the standard moving plane argument used in the proof of Theorem 3.2 can be applied to (7.45) for w(y) along the yi direction; the behavior of w(y) at y = 0 is' irrelevant for the application of this procedure in directions perpendicular to the yi-axis. It follows that w(y) is symmetric about the plane j/2 = 0. As said before, we can replace the 7/2 direction by any direction perpendicular to the yi-axis. Hence w = w(yi, \y'\). By (7.43) and (7.44), we obtain that u = u(xi, \x'\). Going back to the unshifted coordinate system, this implies u(x) = u(x), as long as xj = ~x\. In other words, u = u(xi). UN = 2, the only change in the above argument is that, instead of (7.44), we define w(y) = u(x) = u(—e\ + y/\y\2) and then w(y) satisfies (7.45) with a = 0. The same reasoning then shows u = u(xi). Therefore, in every situation we reduce to the N — 1 case: The problem -u"(t)

= up(t), t > 0, u(0) = 0

(7.46)

has a positive solution. We note that u(t) is a concave function. It cannot increase forever, for if £1 > 0 and u'(ti) > 0, then, u{t) = u{ti) + (t- ti)u'(ti)

+

(t-

s)u"(s)ds,

Jti and if u(t) is increasing for t > ti we obtain by (7.46), for t > t±, 0 < u(t) = u(ti) + (t- ti)u'(ii) -

(t-

s)up(s)ds

Jt\

< u(h) + (t- ti)u'(*i) - (l/2)(t -

hfu'ih),

and it follows

Hence u has a maximum at some point t = t\ and then decreases. But then the concavity of u implies that it reaches zero at some finite point t = t2 < +00. This contradiction shows that (7.42) cannot have a positive solution. D

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Remark 7.34 A simple adaption of the last part of the above proof shows that Theorem 7.31 holds for N = 1. Theorem 7.32 is due to Gidas and Spruck [Gidas-Spruck(1981)]; our proof here follows the original one in [Gidas-Spruck(1981)j. Note that in the proofs of Theorems 7.33 and 7.30, the condition p > 1 is not used; the conclusions in these theorems hold if 0
u > 0 in RN, supu < +oo,

(7.47)

where h is a nondecreasing continuous function satisfying h(0) = 0, h(t) is strictly increasing for t > 0,

lim h(t) = +oo, (7.48) t—>+oo / is C 1 , nondecreasing, satisfying (7.48) and /'(0) = 0. Of particular interests are the special cases f(t) = tp, p > 1 and h(t) = a x t\t\ - or h(t) = (t+)a, a > 0, where t+ = max{i,0}. T h e o r e m 7.37 Suppose that f and h satisfy the above conditions, and u G C1(RN) satisfies (7.47) in the weak sense. Then u = 0. Proof. We use a moving plane argument. As usual, we write, for any real number A, Tx = {x e RN : xx = A}, S A = {x € RN : xx < A}, xx = (x$,x2,...,xN)

= (2A-xi,x 2 ,...,a;jv) ) ux(x)

So xx is the reflection of x in the hyperplane T\.

=u(xx).

Symmetry

and Liouville Type Results over Half and Entire Spaces

155

By the strong maximum principle, any solution of (7.47) is either identically zero or strictly positive in RN. Let us assume that (7.47) has a positive solution u. We will make use of the family of functions w\(x) = u(xx) - u(x) to show that it is nondecreasing in x\; a contradiction then easily follows. For clarity, our proof below is divided into several steps with each step proving a particular claim. We should note that w\ is in general only C 1 under our assumptions. In the following, Aw\(x), etc., should be understood as in Bony's maximum principle (see Theorem 2.10). Claim 1: For x £ Smin{A,o}, wx (x) < 0 implies Awx (x) < 0. Indeed, for such x, xx < min{A,0} < 0. If x^ > 0 then h(xx) > 0 > h(xi). It follows that Awx(x)

- h(x$)f(u(xx))

= h{Xl)f(xi(x))

< 0.

If xx < 0, then h(x{) < h(xx) < 0 and so A«A(X)

= h(Xl)f(u(x)) < h(xx)[f(u(x))

h(xx)f(u(xx))

-

f(u(xx))}.

By assumption, u(x) > u(xx) and hence f(u(x)) — f(u(xx)) h{xx) < 0, it follows that Awx(x) < 0. This proves Claim 1. Claim 2: If A < 0, then w$x) > 0 for every x GT,\. Assume by way of contradiction that for some A < 0,

> 0. As

inf wx{x) < 0. xe'Sx

(7.49)

Define gx(x) = ln(A + 3 - an) + Sf =2 ln[(A + 3 - xx)2 + x% x G

EA+L

A direct calculation shows that the positive function g\{x) satisfies Agx(x) = -(A + 3 - m ) - 2 < 0 in E A + 1 . Let w\(x)

=w\(x)/g\(x).

Since g\{x) —> +00 as \x\ —> +00 in £.\+i, by (7.49), m := inf W\(x) < 0,

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and m is achieved at some io £ S^. Therefore Vwx(xo)

= 0, Aw A (zo) > 0 .

On the other hand, by Claim 1, 0 <

AWA(^O)

< 0 and hence

g\(x0)Aw\(x0)

= Awx(x0)

- 2Vgx(x0)

•

VWA(I0) 2

= Awx(x0) + m(X + 3 - x$-

-

wx(x0)Agx(x0)

< 0.

This contradiction proves Claim 2. Let us define A* = sup{A G R : wM(x) > 0 in EM for every fi < A}. By Claim 2, we find A„ > 0. Claim 3: A, = oo. Otherwise, A* < +oo, and we have a decreasing sequence A.,- —> A* such that inf wXj (x) < 0.

(7.50)

Let M = supflN u(x), CM = sup t e ( 0 > M ] f'(t) and h = /i(A* + 1)C M Then choose small 5 > 0, r\ > 0 such that

^

^ ^ ™ ^ ^ + 1x1 + 1)3 fOT ^ - ^

^-51)

For e G (0, S) sufficiently small (to be specified later) and fixed large k such that Xk < A* + e, we denote A :— Xk and define, for x G £ A , — / x »AI

, ,

WA(I)

= —TV » » 5(x)

x

:= x

A-xi

+ i n • b + X — xi

We observe that 2 > g(x) > 1, Aff(x) =

ff"(:n)

= ^ , 7 ^ , 3 in E A , (o + A - ^ i ) - 3

and 2o' Awx + -^-dXlwx 9

o" 1 o" + ?-wx = -Awx - ~wx. 2g g 2g

(7.52)

Symmetry

and Liouville Type Results over Half and Entire Spaces

157

By (7.50), the set D:={x£Y,x:

wx(x) < 0}

is nonempty. We want to show that if e is chosen properly, 2a' a" Awx + —dXlwx + ?-wx < 0 in D. (7.53) 9 2g If xo £ D is such that Awx(xo) < 0, then (7.53) at XQ follows immediately from (7.52). If x 0 — {x°, •••,x°N) € D and Awx(x0) > 0, then due to wx(xo) < 0, the conclusion in Claim 1 implies that x° > 0. We have two possibilities: ( a ) i ? < \*-5, (b)i?e[A»-J,A]. Consider case (a) first. Since 0 < u(x) < M, standard local estimates give a constant Cx such that |Z?u(:r)| < Cx whenever x lies in the strip 0 < x\ < 2A. Since wx,(xo)

> 0 and wx{xo) < 0, we have w(4*) >u(x0)>u(x£).

(7.54)

Due to x° > 0, we have h(x°) > 0, A > A* > x\ + S > 0, (a;?)A > 0. Moreover, fro" -XO\

= 2(A-A*| < 2 e .

Therefore, 0 > -Awx(xQ)

= / l [(x°) A ]/[w(4)] -

/I(I;)/[U(I0)]

>(/ l [(x?) A ]-/ l K))/K4*)] + ^[K)A](/[«(^o)]-/K4*)]) > [A(A„) - fc(A» - S)]f[u(x£')]

- J»(2A„ + l)C M [w(a:o*) - u(x%)]

> [/i(A,) - h{\* - 6)]f[u{x%')] - h{2K + It follows that j?r i \.w ^ ^(2A* +

1)CMCX

l)CMCx(2e).

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which implies U(XQ*) < 77 if e is small enough. We suppose that e is so fixed. Then, due to (7.51) and (7.54), -Awx(xo)

>

fc(*i)(/[u(zo)]

-f[u(x0)})

6/2

w x ^ h(^*)T7\—r~m—r~r^ 'fc(A,+T)(A* + l) *( °)

>

b2W$x0),

where

62 :=

ixrnr9{xo)

* (5+x-x°)* = -(vajA^o),

due to 0 < 5 + A - x\ < A* + 26, 1 < g(xo) < 2. Substituting these inequalities into (7.52) we obtain (7.53). Next we consider case (b), where 2;° € [A* — 6, A]. In such a case,

-Awx(x0)

> &(*?)(/[«(*$)] " /[«(*(>)]) > /i(A* + I ) C M ^ A ( ^ O ) bl

— 1

\

ff(zo) W\(XQ). By (7.51), -(V2)A,(,o) =

6 {6 + x

_xo)3

> (^3 > * > Vsto)-

Again (7.53) follows. Therefore (7.53) always holds. We now use a comparison argument to show that (7.53) leads to a contradiction. To this end, we consider the auxiliary problem Av + a(xi)vXl

+ b(xi)v = 0 in BR(0),

V\9BR(0)

where a = — Vxi < A, a = ——— vxi > A; 9 5(A)

& = f Van < A, 6 = f ^ Va* > A. 2g 2g{$

= -M,

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159

Since b < 0, this problem has a unique solution VR and by the m a x i m u m principle, — M < VR < 0 in BR(0). We must have w\ > VR in BR(0) n T,\ for otherwise, by (7.53) and the maximum principle, we easily deduce w\ > vR in {x e BR(0)

n E A : wx(x)

< vR(x)}

C D,

which is clearly a contradiction. T h e maximum principle also infers t h a t VRX < VR2 in BR1(0) if i?i < R2. Therefore v := lim^-joo vR is well-defined. By s t a n d a r d regularity theory, v is a solution to Av + a(x!)vXl

+ b{xx)v = 0 in RN.

(7.55)

Moreover, we have vR < v in BR(0),

WX > v in S A and -M

< v < 0 in i ? ^ .

If v0 is any solution of (7.55) satisfying —M < vo < 0, then a simple application of the maximum principle shows t h a t VR < VQ, which implies v
(7.56)

Since — M < v < 0, we can easily show t h a t v = 0. Suppose this is not true. T h e n the strong maximum principle implies t h a t v < 0. We first observe t h a t v must be monotone. Indeed, if it has a negative local minimum at to, then v"{tQ) + a(tQ)v'(t0)

+ b(to)v(to)

> 0,

which is a contradiction. This implies t h a t v(x\) can have a t most one local maximum, and both u(+oo) and u(—oo) exist. We must have v(+oo) = 0 for otherwise, the left side of (7.56) becomes positive for large x\. This implies t h a t v is increasing. On the other hand, for x\ < A, we have {g2v')'

= -(l/2)g"gv

< 0.

Therefore, for t < to < A, we have g2(t)v\t)

> g2(t0)v'(t0)

> 0, v'(t) > (l/4)t/(*„)•

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This is impossible since v(—oo) > -M. Therefore v = 0 and w\ > 0 in TJ\. But this is in contradiction with D ^ 0. The proof of Claim 3 is finally complete. We are now ready to complete the proof. By Claim 3, we find u(x) < w(xA), Vx G £ A , VA G R1. Therefore, u(x) = u(xi,x) is nondecreasing in x\. Let us define, for each a > 0, na = {x= (x!,x) G RN : a < xx < a + 1, \x\ < 1}. Clearly the first eigenvalue /J,I of the problem - A u = /xu in fiff, w|anff = 0 is independent of a. Since M is a positive solution of (7.47) and u is nondecreasing in x\, we have - A u = h(xi)f(u)

> h((x)m0u, Vx e

fia,

( 7 - 57 )

where m

o = A ^ > 0, £o = M

min

u(x) > 0.

xi=0,|a:|
Since u > 0 in i ? w , (7.57) implies /xx > h(a)mo- It follows that limCT_+00/i(cr) < ni/m0

< +oo,

a contradiction to our assumption. Therefore (7.47) has only the trivial solution u = 0. This completes the proof of our theorem. • Remark 7.38 Theorem 7.37 was first proved in [Du-Li(2005)] for a less general case which nevertheless covers the examples mentioned before the theorem. The proof there follows that of C.S. Lin in [Lin(1998)] for the special case h(x\) = x™, m an odd integer, and f(u) = u( JV+2 )/( JV_2 ) (N > 3). Remark 7.39 Our proof given above used an idea of Polacik and Quittner [Polacik-Quittner(2004)], where they have proved a more general result, namely, under the assumptions on h and / as in Theorem 7.37, if «(x, t) is defined for all t G R1 and x G RN, and is a bounded nonnegative solution of ut - Au = /i(xi)/(u), xeRN,te then u = 0.

R1,

Symmetry

and Liouville Type Results over Half and Entire Spaces

161

If h(x\) in (7.47) is non-positive, we have a similar Liouville theorem, which holds even if the equality is replaced by an inequality. This result improves the main result in [Keller(1957)] and is taken from [Du-Li(2005)]. Theorem 7.40

Suppose that h E C(R}) satisfies

h(t) < 0 for all t, h(t) <0fort<0

and Irm t _ ( _ 00 /i(t) < 0.

(7.58)

Let f(u) be a C1 function of u for u G [0, oo) that is positive and nondecreasing, and satisfies /(0) = 0, f™ F(u)-ll2du Then the only function u € ^(R -Au

< h(xi)f{u),

< oo, where F(u) = /„" f(s)ds.

(7.59)

) satisfying (in the weak sense) u > 0 in RN, supu < +oo

(7.60)

is u = 0. Proof.

By (7.58), we can find h E C^-R1) such that

h{t) > h(t) Vt G R1; h{t) < 0 W < 0; h(t) = 0 Vt > 0; Rmt_>_oo^(t) < 0. Suppose by way of contradiction that there exists some u» G C^-R-^) satisfying (7.60) with M := supfljv u* > 0. For Bn := {x G RN : \x\ < n}, we consider the problem - A u = ~h(xi)f(u) in Bn, u\dBn = M.

(7.61)

Clearly u — 0 and u = M are lower and upper solutions to (7.61). Therefore it has at least one solution satisfying 0 < u < M. Since h < 0 and f(u) is nondecreasing in u, a standard comparison argument shows that (7.61) has a unique solution. Let us denote this solution by un. The same comparison argument also shows that un > un+i > u* on Bn for all n > 1. Therefore u(x) := linin^oo un{x) is well-defined for every x G RN, and u > um in RN. Moreover, a standard regularity consideration shows that u G Cl(£(R ) (Vg > 1) is a weak solution of -Aw = h(a;i)/(«) in i ? ^ .

(7.62)

We claim that w is the maximal solution of (7.62) among all nonnegative solutions of this equation satisfying u < M in RN. Indeed, if u is such a solution of (7.62), then by the comparison argument used above, it follows from U\QBU < M = un\gBn that u < un in Bn and hence u < u in RN.

Maximum

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Applications

For any given y0 = {0,y%, ...,y%) G RN, uyo(x) := u(x + y0) clearly solves (7.62). Therefore u(x + y0) < u(x). It follows t h a t u is a function of xi only, namely, u(x) = satisfies -" = h{x{)f{(j)),

0 < < M in R1.

In particular, " = 0 and 0 is a bounded function in [0,00). Therefore, we necessarily have <j>{x{) = m on [0, 00) with 0 < m < M. We now find t h a t 4> is the unique solution of the initial value problem -" = HxM(),

(0) = m,
We must have m > 0 for otherwise the uniqueness of the initial value problem implies <j> = 0 on R1, contradicting <> / = M > M*. AS 4>" = —h{x\)f{4>) > 0, we find that '(0) = 0 for x\ < 0 and hence {x\) > m for x\ < 0. By our choice of h, there exists a > 0 such that h(x{) < — a for xi < - 1 . Thus <£" > ^/(<£), (/> > m for xi < - 1 . It follows that ([ [^»'(-l)] 2 /2 > -aF(<j){-l)),

oFM-l)) Vii < - 1 .

From this we deduce, with i*i(u) = F ( u ) — F(4>(—1)),

J(-2)

J(-2)

>{2a)1''2{-2-x1), It follows t h a t JJ)/_2) Fi(u)~l/2du

VXl < - 2 .

= 00. One easily sees t h a t this implies

oo

/ a contradiction to (7.59).

F ( u ) - 1 / 2 ^ = 00, •

Appendix A

Basic Theory of Elliptic Equations

For convenient reference of the reader, we include here a brief review of the classical elliptic theory for second order partial differential equations. A.l

Schauder theory for elliptic equations

Suppose that fi is a bounded domain in RN with boundary d£l. Consider the following problem Lu = / in ft, Bu = 4> on dfl.

(A.l)

Here Lu = alJ (x)Dijii + bl(x)Diii + c(x)u, DiU denotes uXi, DijU denotes uXiXj, and the summation convention is understood. We assume that the coefficient matrix \ai:>{x)] is symmetric and positive definite at every x € Q., and moreover, there exist constants A > A > 0 such that

m2<^{x)^<m\2 for all £ G RN and all a: G Q. Under these assumptions, L is known as a strongly uniformly elliptic operator. A typical such operator is the Laplacian A: AM = YlDuu. The boundary operator Bu is given by Bu = u, (Dirichlet boundary operator), 163

Maximum

164

Principles

and

Applications

or Bu = Duu, (Neumann boundary operator), or Bu — a(x)u + D„u, a(x) > 0, ^ 0, (Robin boundary operator), where Duu denotes the directional derivative of u in the direction of the unit outward normal v = v(x) of dQ.. The existence of classical solutions to (A.l) can be established by the well-known Schauder theory, which is based on the use of Holder spaces. For a constant a G (0,1], the Holder space Ca(fl) consists of continuous functions u(x) on Q which satisfy \u(x)

r i

[u]a :=

sup

'

;

— u(y)\

_

/f1

< oo.

If k is a nonnegative integer, then the Holder space Ck'a(Q) consists of functions with continuous partial derivatives up to A;-th order, and all the fc-th order partial derivatives are in C"*(fi); its norm is given by l|w||fcQ:= Here (3 = (/?i,..., index,

/?JV),

>

maxmax \D^u\ +

max\D^u}a.

ft = nonnegative integer, with \(3\ = Eft, is a multi-

dxl1

• • • dx%N '

Ck'a(Q) is a Banach space. To avoid possible confusion later, let us remark that Du will be used for the gradient of u: Du = (D\u, • • • ,

DNU).

We say Q is Ck'a, or more accurately its boundary dfl is Ck,a, if dti is a (N — l)-dimensional Cfc'a-manifold, that is, near each point on <9fi, there is a parameterization of dQ, with every function involved in the parameterization belonging to a space Ck G C2'a(dCl).

Basic Theory of Elliptic

Equations

165

Then the Dirichlet problem Lu = / in CI, u = on dCl, has a unique solution lying in C2'a(Cl). Moreover, there exists a constant C independent of u, f and
\\uh,a
+ \\f\\a).

Theorem A.2([Gilbarg-Trudinger] Theorem 6.31) Let L be strictly uniformly elliptic in a bounded domain CI, with c < 0, ^ 0, and let f and the coefficients of L belong to Ca(Cl). Suppose that Cl is C2'a andcf) £ C1,a(dCl). Then the Neumann problem Lu = f in Cl, Dvu = <j> on dCl, has a unique solution lying in C2'a(CY). Moreover, there exists a constant C independent of u, f and such that ||u||2ia on dCl, has a unique solution lying in C2'a(fl). Moreover, there exists a constant C independent of u, f and <j> such that

))uh,a
+

\\f\\a).

Let us remark that the conditions on c in the previous theorems are sufficient but not necessary. A necessary and sufficient condition involves the eigenvalue problem. If we denote by B the boundary operator which can be Dirichlet, or Neumann, or Robin type, then the following theorem holds. Theorem A.4 Let L be strictly uniformly elliptic in a bounded domain CI, with coefficients belonging to Ca(Cl). Suppose that CI is C2'a and a e CX'a(dCl) when B is of Robin type. Then the eigenvalue problem Lu + Xu = 0 in Cl, Bu = 0 on dCl,

(A.2)

166

Maximum

Principles

and

Applications

has a nonzero solution (lying in C2'a(fl)) if and only if A 6 S, where £ is a sequence of complex numbers {A n }, called eigenvalues, with Xi real, and satisfying \\ < inffc>2 Re(Xk). Moreover, complex eigenvalues come in pairs of the form r\ + £i, r\ — £i, and by a nonzero solution of (A.2) with A = r\ + £i, we understand a function of the form u = v + wi, where v and w are real valued functions. Clearly v — wi is a nonzero solution with \ = n-&. It is evident that the eigenvalue set E depends on the coefficients of L, in particular c. We can now state a necessary and sufficient condition on the solvability of (A.l). Theorem A.5 In Theorems A.l, A.2 and A.3, the condition on c can be replaced by 0 $ £. Conversely, if 0 £ S, then there exist f and 0. Therefore these theorems are consequences of Theorem A.5. The smoothness requirements in the above theorems are more than necessary for the existence and uniqueness of a solution to (A.l). We refer to Theorem 6.13 in [Gilbarg-Trudinger] for more details.

A. 2

Sobolev spaces

Let Q, be a bounded domain in RN, u a locally integrable function in ft and /? any multi-index. Then a locally integrable function v is called the /?-th weak derivative of u if it satisfies f vdx = (-1) I/3 I /

Jn

Jo.

uD^^ds

for all cf> G C'0 (Cl). Here CQ(Q) denotes all functions with continuous partial derivatives of order up to k, and with compact supports in fi. We write v = D@u and note that D^u is uniquely determined up to sets of measure zero. Clearly Ck(Q.) c Wk(Q). If D. contains the origin, then it is easily checked that u0(x) = l / | x | a belongs to Wk(Q.) provided that k + a < N. For p > 1 and k a nonnegative integer, we let Wk*{n)

= { « e Wk(Q) : Dpu G Lp{Q), V|/3| < k}.

Basic Theory of Elliptic

Equations

167

A norm in this space is usually given by

Hw.p(n) : = ( / J2 \ D l 3 u \ P d x ) ]

\u\\k,p;Q

I/P

" '°'l/3|
An equivalent norm would be

||U|| := £

\\D^u\\LP{n).

\0\
Under either norm, W ' (Q) is a Banach space. Moreover, it is a Hilbert space if p = 2. These spaces are usually called Sobolev spaces. Functions in a Sobolev space need not be continuous, as the example uo(x) = l/|a;| a shows. However, there is a certain continuity requirement for a function to belong in a Sobolev space. Theorem A.6([Ziemer(1989)] Theorem 2.1.4) Suppose u £ Lp{Cl), p > 1. Then u £ W 1,p (fi) if and only ifu has a representative u* that is absolutely continuous on almost all line segments in Q parallel to the coordinate axes and whose (classical) partial derivatives belong to Lp(fl). A function u : fl —•> R1 is said to be differentiable at x e CI if there exists a € RN such that Um

y->x

\u(y) - u(x) - a • (y - x)\ \y — x\

= Q

It is easy to check that a, if exists, is unique. It is usually denoted by Du(x), called the gradient of u at x. Theorem A.7([Evans(1998)] Chapter 5, Theorem 5) Assume that u 6 W1,p(fl') for any fl' whose closure is contained in ft, and p > N. Then u has a representative u* that is differentiable a.e. in fi, and the gradient Du* equals the weak gradient Du a.e. in Cl. The positive and negative parts of a function u are defined by u+ = max{u, 0}, u~ — min{u, 0}. Clearly u = u+ + u~ and |u| = u+ — u~. We have the following result. Theorem A.8 ([Ziemer(1989)] Corollary 2.1.8) then u+ ,u~ ,\u\ € Wl'p(fl), and Du

Du+-[

\0

i u>0

f

ifu<0.

Ifu

6 Wx'p(n),

p > 1,

Maximum

168

Principles

and

[Du D\u\ =Du+

Applications

ifu
The above theorem is a special case of the following result. Theorem A.9 ([Ziemer(1989)] Theorem 2.1.11) Let f : R1 -> R1 be a Lipschitz function and u G Wl'p(Cl),p > 1. / / the composition f o u G Lp(£l), then fou€ W 1,p (fi) and for a.e. x G Q, D(f ou)(x) =

f'(u(x))Du(x).

It is useful to know what effect a change of variables has on a Sobolev function. T : RN —> Ji^ is called a bi-Lipschitzian map if it is 1-1 and there exists some constant M such that T and T~x satisfy \T(x) - T{y)\ < M\x - y\, IT-^x)

- T-l(y)\

< M\x - y\, Vx, y € RN.

It is easily seen that Theorem A. 7 implies the following classical result of Rademacher: A Lipschitz map T : RN —> RN is differentiable at almost every points in RN; that is, for a.e. x G RN, there is a linear map dT(x) : RN -> RN such that lim

\T(y)-T(x)-dT(x)(y-x)\

y—>x

= Q

\y — a; I

The following result is concerned with a bi-Lipschitzian change of coordinates for Sobolev functions. Theorem A.10 ([Ziemer(1989)] Theorem 2.2.2) Let T : RN -> RN be a bi-Lipschitzian map and u G W1,P(Q,). Then v := u o T G W1^^!') with ft' ^ T - ^ f i ) , and Dv(x) • y = Du{T(x)) • dT{x)y for a.e. x G fl' and all y G RN. The next result shows that Sobolev functions can be approximated by smooth functions. Theorem A . l l

The subspace C°°(Q) n Wk'P(Q) is dense in Wk
Basic Theory of Elliptic

Equations

169

Clearly C£°(ft) c Wk'p(Q) for any nonnegative integer A; and constant p > 1. The closure of C0°°(ft) in Wk X2, if X\ C X2 and the injection mapping from X\ to X2 is continuous. The imbedding is said to be compact, written X\ «—><—> X2, if the injection mapping is further compact. Theorem A.13 W^'P(Q) ^ LNp^N-P\n)

ifl
W01>p(ft) <-• C A (ft), A = 1 - —, Moreover, there exists a constant C = C(N,p)

\\U\\NP/(N-P)

(A.3)

ifP>N. such that, for any u £

< C\\Du\\p for p < N;

(A.4)

sup |u| < Clfil^^-^'HDuHp, [u]x < C[l + (diamft)A]||£>w||p for p > N. a Remark A.14 The imbedding (A.3) and the inequality (A.4) are true also when ft is unbounded. If p — N, then WQ'P <—> L^(ft), where <j> = el*' — 1 and L^ is the Orlicz space with defining function <j>. By iterating the result of Theorem A.13 k times we arrive at an extension to the space WQ ' p (ft). Corollary A. 15 Wk'p(n)

^ LNp/iN~kp\n)

ifkp < N;

Maximum

170

wk,P

^

Principles

and

m

>x(Q) ifO<m
C

Applications

<m + l,X = k P

m. P

Remark A.16 In general, W0 ' p (fi) in Corollary A.15 cannot be replaced by Wk'p(Q). However, this replacement can be made for a large class of domains Q, which includes domains with Lipschitz continuous boundaries. More generally, this replacement can be made if fi satisfies a uniform interior cone condition: There exists a fixed cone K = KQ such that each x £ fi is the vertex of a cone K(x) C Q. which is congruent to K. Theorem A. 17 W£'p(fl) *-*<-> Lq(fl) ifkp
— C m ^(Q) ifO<m
N P

Np/(N

- kp);

Remark A.18 Theorem A.17 is valid when W0'p(fl) Wk'p(Q) if Q has Lipschitz continuous boundary.

A.3

N

<m + l,n
P

m.

is replaced by

Weak solutions of elliptic equations

We consider linear elliptic operators L having principal part in divergence form Lu = Di(aij(x)DjU

+ ^(x)^) + ci(x)Diu + d(x)u

whose coefficients a11,b%,cl,d (i,j = 1,...,N) are measurable functions on a domain fl c RN. If all the highest order coefficients a i j are C 1 functions, then the above L can be easily rewritten in nondivergence form; conversely, an operator in nondivergence form Lu = alj(x)DijU + b\x)DiU + c(x)u can be rewritten in divergence form if all the
Basic Theory of Elliptic

Equations

171

integrable, then, u is said to satisfy Lu = 0 (> 0, < 0) respectively in fl in the weak sense, if / [aijDjU + VujDiV - (c'DiU + du)v]dx = 0 (< 0, > 0)

Jo.

for all nonnegative function v e CQ?(Q). Provided the coefficients of L are locally integrable, it follows from the divergence theorem that a function u G C2{fl) satisfies Lu = 0 (> 0, < 0) in the classical sense also satisfies these relations in the weak sense. Moreover, if the coefficients a%\ bz have locally integrable derivatives, then a generalized solution u e C2 (fl) is also a classical solution. Let / ' , g, i = 1,..., N be locally integrable functions in fl. Then a weakly differentiate function u will be called a weak solution of the inhomogeneous equation Lu = g + Dif in fl,

(A.5)

ifVveC 0 °°(Q), / [aijDjU + VujDiV - (JDiU + du)v\dx = f (fDiV

Jn

Ja

- gv)dx.

(A.6)

Note that in (A.6), the test function v can also be taken from CQ(Q), the space of functions with continuous first order partial derivatives and compact supports in fl, since Co°(fl) is dense in this space. We would like to remark that CQ(Q) should be distinguished from CQ(CI), the latter usually being used to denote the space of functions in C1 (fl) that take value 0 on dfl. We now assume that ai^,bi,ci,dGL'x(fl), i,j = l,...,N; ij x A 2 a ( )titj > IC| for some A > 0 and all x £ fl, £ 6 RN;

(A.7)

(dv - VDiVJdx < 0 for all nonnegative v e CQ(£1).

(A.8)

Ja >a

T h e o r e m A. 19 ([Gilbarg-Trudinger] Theorem 8.3) Let the operator L satisfy (A.7) and (A.8). Then for any given (j> S W1'2(fl) and g,fl € L2(fi), i = 1, ...,N, the generalized Dirichlet problem Lu = g + Dif1 in fl, u = <j> on dfl has a unique weak solution u e

W1,2(fl).

172

Maximum

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and

Applications

Here u =
€ Wo' (fi). We can also consider weak solutions of (A.5) with other boundary conditions. To match L in the integral formulation, the corresponding Neumann and Robin boundary operators should have the following form Nu := alJ\x)ViD'jU + bl{x)viU + o~(x)u, where v = (I>\,...,VN) denotes the outward unit normal of dfl which we assume is C 1 . For a, £ L2(dCl), u £ W1'2(Q) is called a weak solution to Lu = Dif

+ g in f2, Nu = 4> on dQ,

(A.9)

if Jn[aljDju + blu)DiV — {clDiU + du)v]dx = fn(fDiV - gv)dx + faa(4> - fvi - au)vds W £ W 1 ' 2 ^ ) .

. (

}

Since u £ W 1,2 (fi) is only determined up to a set of measure 0, the term u appearing in the integral over 80, does not make sense, as dQ, has measure 0 in RN. Here we need the notion of trace to properly understand this case. T h e o r e m A.20([Evans(1998)] Section 5.5 Theorem 1) If Q is a bounded domain with C1 boundary dQ, and 1 2(fi), where the constant C depends only on p and CI. We call Tu the trace of u on dVi. It can be shown that u £ WQ'P(£1) if and only if u £ W 1,p (fi) and has trace 0 on dQ; see [Evans(1998)] Section 5.5 Theorem 2. In fact, the trace can be defined and this fact remains valid if dCl is only Lipschitz continuous (see [Ziemer(1989)] page 190). More generally, if fl is a bounded domain in RN with Cm boundary d£l, m > 1, and u £ W m -P(fi), then Tu is a bounded operator from Wm-P(fi) to Lq(dQ) for g £ [p, NZmp ] when mp < iV; for 9 £ [p, 00) if mp > iV. See Theorem 5.22 and Theorem 7.53 in [Adams(1975)] for more details. We now see that (A. 10) makes sense if we understand u as its trace on dfl in the last integral there.

Basic Theory of Elliptic Equations

173

If we let B denote either t h e boundary operator Bu = u or Bu — Nu with N as given above, then t h e following result holds. T h e o r e m A . 2 1 Let the operator L satisfy eigenvalue problem

(A.7) and (A.8).

Lu + Xu = 0 inQ, Bu = 0 on dft, u £

Then the

Wh2(n)

has a countable set of eigenvalues £ = {A,} in the complex plan, with the first one real and complex eigenvalues arise in conjugate pairs, moreover, Ai < Re(Xi) for all i>2, and |A;| —> oo as i —> oo. T h e o r e m A . 2 2 Let the operator L satisfy (A.7) and (A.8) and suppose 0 ^ £ . Then the problem Lu = Dif1 + g in fi, Bu = on dQ has a unique weak solution in Wl'2(Q) for any given g,fl £ L2{il),i 1,..., JV and any given 4> £ Wl,2(Q) (when Bu = u) or (j> £ L2{dQ) Bu = Nu).

= (when

R e m a r k A . 2 3 In Theorem A.22, condition (A.7) can b e relaxed. In [Ladyzhenskaya-Ural'tseva(1968)], similar results are obtained by requiring b\ d £ Lq(£l), c £ I/*/ 2 (ft), where q > N (and one c a n choose q = N when

N>3). For weak solutions, there is a theory on interior a n d global regularity. We list here two interior regularity results. T h e o r e m A.24([Gilbarg-Trudinger] Theorem 8.8) Let u £ W 1 , 2 ( f i ) be a weak solution of Lu = f in Q where f £ L2(fl), L satisfies (A.7) and furthermore, al:>,bl, i,j = 1, ...,N are uniformly Lipschitz continuous in Q.. Then for any subdomain ft'ccfl (meaning the closure ofQ' is contained 22 in £1), we have u £ W ' (fl') and H l i y ^ ( f i ' ) < C(\\u\)wi,2{n) where C depends only on L and

+ ||/||i2(n)),

d(Cl',dfl).

T h e o r e m A . 2 5 ([Gilbarg-Trudinger] Theorem 8.24) Let the operator L satisfy (A.7), and suppose that p e Li(Sl),i = 1,...,N, g £ L'"'2^) for some q > N. Then, if u £ W1'2^) satisfies Lu = Dip + g in Q, in the weak sense, we have for any ft' C C ft the estimate l|w||c°.°(fi') < 0(\\u\\L2(n)

+ S||/l||L,(n) + H s H i ^ n ) ) .

Maximum

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Principles

and

Applications

where C depends only on L and d(Cl',dCl). A.4

Lp theory of elliptic equations

Parallel to the Schauder theory there is an Lp theory for elliptic operators in the general form Lu = alj(x)DijU + bl{x)Diu + c(x)u. In this theory one looks for a solution u G W2'p(Cl) which satisfies Lu = f a.e. in Cl, together with suitable boundary conditions. Such solutions are usually called strong solutions. While this theory requires more smoothness of the solution (belonging to W2'p(Cl)) than the weak solution theory (which only requires u G W 1,2 (fi)), the operator L now needs not be in divergence form. We have the following interior estimate, where by Wt '£(Cl) we mean the space of functions which belong to Wk'p(Cl') for every Cl' CC Cl. Theorem A.26([Gilbarg-Trudinger] Theorem 9.11) Let Cl be an open set in RN and u G Wlt£(Cl) n Lp(Cl), 1 m2

V£ G i ^ ,

\aH\, \b% |c| < A,

where i,j = 1,..., N. Then for any domain Cl' CC Cl,

IMIw2,P(n') < C(l|w||Lp(n) + Wfhnci)), where C depends on N,p,X,A,Cl',Cl Cl'.

and the moduli of continuity of a,J' on

Next is a local boundary estimate result. Theorem A.27([Gilbarg-Trudinger] Theorem 9.13) Let Cl be a domain with a C2 boundary portion T C dCl. Let u G W2'p(Cl), 1 < p < oo, be a solution of Lu = f in Cl with u = 0 onT, in the sense ofW1,p(Cl) (i.e., u is the limit in W1,p(Cl) of a sequence of functions in Cx(f2) vanishing near T), where L satisfies the conditions in Theorem A.26 and furthermore, aij G C(Cl U T). Then for any domain Cl' CCClUT, \M\W^P(Q')

< C(\\u\\LP{n)

+

\\f\\Lr(n)),

Basic Theory of Elliptic

Equations

175

where C depends on iV,p, A, A,T, ft',ft and the moduli of continuity of a' J on ft'. These estimates lead to the following existence and uniqueness theorem for the Dirichlet problem. Theorem A.28([Gilbarg-Trudinger] Theorem 9.15) Let Q be a bounded C2 domain, and let the operator L satisfy the conditions in Theorem A. 26 and furthermore, a IJ G C(ft), i,j = 1,...,N, and c < 0. Then, for any f £ L p (ft) and 4> £ W2>p(tt), with 1 £ W0'p(Cl) has a unique solution u £ W2,P(Q). To consider other boundary value problems, we now follow the approach in [Agmon-Douglis-Nirenberg(1959)]. Suppose that ft has C2 boundary <9ft. Then by the trace theorem every u £ Wk'p(Cl) with fc>l,l which are the boundary values (i.e., the trace on dft) of functions v belonging to Wk'p(Cl). In this space, we introduce the norm

Let Bu — Dvu+a{x)u global estimate.

be the boundary operator. We have the following

Theorem A.29([Agmon-Douglis-Nirenberg(1959)] Theorem 15.2) Let ft be a bounded domain with C2 boundary dfl. Suppose that the operator L has C(ft) coefficients and satisfies for some positive constants A, A,

m2 < aijtej < A|£|2 V ^ i ? w ; a £ Cx(dCl); / £ L p (ft), 4> £ H^p.

Then, for any u e W2>P(Q) satisfying

Lu = f in CI, Bu = \\*,p +

\\U\\LP{CI))>

where C depends on L and ft only. Remark A.30 (i) In Theorem A.29, we do not need all the coefficients of L to be continuous on ft; it is enough if ali £ C(ft) and b\c £ L°°(ft). (ii) If the boundary operator is of Dirichlet type: Bu = u, then Theorem A.29 still holds if we require <j> £ H^p and replace ||<£||iiP by \\4>\\2,P m estimate.

tne

176

Maximum

Principles

and

Applications

The following two theorems are the analogues of Theorems A.4 and A.5. Theorem A.31 Let Q be a bounded domain with C 2 boundary <9Q. Suppose that the operator L has C(Cl) coefficients and satisfies for some positive constants A, A,

m2

< aijtej

< A|£|2 V£ e RN;

and that the boundary operator has the form Bu = u or Bu = Dvu + a(x)u with a £ C1 (dCl). Then the eigenvalue problem Lu + \u = 0inn,

Bu = 0 ondto,

(A.ll)

has a nonzero solution (lying in W2'P(Q), 1 ) if and only if A G E, where £ is a sequence of complex numbers {A n } ; called eigenvalues, with Ai real, and satisfying X\ < inffc>2 Re(Xk). Moreover, complex eigenvalues come in pairs of the form rj + £i, r) — £i, and by a nonzero solution of (A.ll) with A = r\ + £i, we understand a function of the form u = v + wi, where v and w are real valued functions. Clearly v — wi is a nonzero solution with \ = ri-£i. Theorem A.32 Let the conditions of Theorem A.31 be satisfied and 0 ^ E. Then for any f S LP(Q,), 4> £ H*p when Bu = Dv + au, and G H^p when Bu = u, the problem Lu = f in CI, Bu = on dCl has a unique solution u G W2'P(Q,). Using interior estimates and barrier argument, the Dirichlet problem can be solved with much less smoothness requirements for dfl. We say ft satisfies an exterior cone condition at XQ G dQ if there exists a finite right circular cone V = VXo with vertex XQ such that 0. D VXQ = x0. Theorem A.33([Gilbarg-Trudinger] Theorem 9.30) Let L be as in Theorem A.26 and c < 0. Suppose that Q. satisfies an exterior cone condition at every point on its boundary. Then for any f G Lp(Q),p > N and every 4> G C(dfl), the Dirichlet problem Lu = / in Q., u = <j> on dVl, has a unique solution u G W^p(Cl) n C(ty.

Basic Theory of Elliptic

A.5

Equations

177

Maximum principles for elliptic equations

The maximum principle is an important feature of second order elliptic equations that distinguishes them from equations of higher order and systems of equations. It also plays a vital role in the study of nonlinear problems involving second order elliptic operators. Here we collect some wellknown forms of the maximum principles. A.5.1

The classical maximum

principles

Consider the operator Lu = Aij(x)DijU

+ bl(x)DiU + c(x)u.

We suppose that [a u ] is a symmetric matrix but we do not require any smoothness conditions on the coefficients of L. L is said to be elliptic in a domain Q C RN if there exist A(i), A(:r) > 0 such that \(x)\£\2

< a^{x)Uj

< A(:r)|£| 2 , V£ &RN,x£

Q.

If A/A is bounded in fj, then we call L uniformly elliptic in Cl. If A(x) > Ao > 0 in ft, then L is said to be strongly elliptic in fl. Throughout this subsection, we assume that \b\x)\X(x)

< C 0
l,...,N,x€Sl.

Theorem A.34 (Weak maximum principle, [Gilbarg-Trudinger] Corollary 3.2) Let L be elliptic in the bounded domain fi. Suppose that in tt, u G C 2 ( 0 ) n C ( f i ) satisfies Lw>0(<0), c<0. Then supu < s u p u + (inf u > inf u~). n an n dn Let us recall that u+ = max{u, 0}, u~ = min{u,0}. The strong maximum principle is based on the following lemma, known as the Hopf boundary lemma , where we say that the domain D. satisfies an interior sphere condition at XQ £ dCl if there exists a ball B C f2 with i 0 e dB.

Maximum

178

Principles

and

Applications

Lemma A.35 (Hopf boundary Lemma [Gilbarg-Trudinger] Lemma 3.4) Suppose that L is uniformly elliptic, c = 0 and Lu > 0 in fl, where u G C 2 (fi). Let XQ G dQ be such that (i) u is continuous at XQ; (ii) U{XQ) > u(x) for all x G fi; (Hi) dft satisfies an interior sphere condition at XQ. Then Dvu{xQ) > 0 whenever the directional derivative exists, where v is a unit vector pointing outward ofQ at XQ. Ifc<0 and c/\ is bounded, the same conclusion holds provided u(xo) > 0, and ifu(xo) = 0 the same conclusion holds irrespective of the sign of c. Theorem A.36 (Strong maximum principle [Gilbarg-Trudinger] Theorem 3.5) Let L be uniformly elliptic, c = 0 and Lu > 0 (< 0) in a domain fi (not necessarily bounded). Then if u achieves its maximum (minimum) in the interior of Cl, u is a constant. If c < 0 and c/\ is bounded, then u cannot achieve a non-negative maximum (non-positive minimum) in the interior of Vt unless it is a constant. A.5.2

Maximum principles weak solutions

and

Harnack

inequality

for

The classical weak maximum principle has a natural extension to operators in divergence form: Lu = Di(aij (X)DJU + V^u)

+ ci(x)Diu + d{x)u,

where we assume that (A.7) and (A.8) are satisfied. Theorem A.37 (Weak maximum principle, [Gilbarg-Trudinger] Theorem 8.1) Let u G W 1 - 2 ^ ) satisfy Lu>0(< 0) in Q. Then supu < supw + (inf u > inf u~). n an n an Here snpmu -supan(-u).

:= inf{fc G Rl

: (u - k)+ G W 0 1,2 (ft)};

infanu

=

Theorem A.38 (Strong maximum principle, [Gilbarg-Trudinger] Theorem 8.19) Let u G Wx'2{Vi) satisfy Lu > 0 in ft. Then, if for some ball B c c f i

Basic Theory of Elliptic

Equations

179

we have sup B u — sup a u > 0, the function u must be constant in Q, and equality holds in (A.8) when u ^ 0. Theorem A.39 (Harnack inequality, [Gilbarg-Trudinger] Corollary 8.21) Let the operator L satisfy (A.I) but not necessarily (A.8), and let u £ Wl'2{VL) satisfy Lu = 0, u > 0 in Q. Then for any Q' CC Cl, we have sup it < Cinf u, Q'

"

"'

where C depends only on L, Q and CI'. A.5.3

Maximum principles strong solutions

and

Harnack

inequality

for

Let L be as in subsection A.5.1 and suppose that it is elliptic in Cl. Let det[ali] denote the determinant of the coefficient matrix [alJ'(a;)]. Theorem A.40 (Aleksandrov weak maximum principle, [GilbargTrudinger] Theorem 9.1) Suppose that CI is bounded, c < 0 and |6*|/det[a y ], f/det[aij] Let u e W?£{Cl) satisfy Lu>f

e LN(Cl), i = 1,...,N.

inCl. Then

supu < limx^dQU+ + n

C\\f/det[alJ}\\LN,m,

where C is a constant depending only on N, diamCl and ||6/det[a,-7']||£,w(fj). Theorem A.41 (Strong maximum principle, [Gilbarg-Trudinger] Theorem 9.6) Suppose that L is uniformly elliptic, and |6 l |/A,c/A are bounded in Cl. IfuG Wl(^c (Cl) satisfies Lu > 0 in Cl and c = 0 (c < 0), then u cannot achieve a maximum (nonnegative maximum) in Cl unless it is a constant. Theorem A.42 (Harnack inequality, [Gilbarg-Trudinger] Corollary 9.25) Suppose that L is strictly elliptic with bounded coefficients in the domain Cl, and let the constants 7, n satisfy A / A < 7 , (|6|/A) 2 ,|c|/A
180

Maximum

Principles

and

Applications

Let u G W2'N(£l) satisfy Lu = 0, u > 0 in fi. Then for any ball B2R(V) C fl, we have sup u < C inf u, B BR(V) «fe) where C =

C{N,-y,r1R2).

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Index

bifurcation from zero, 67 from infinity, 67

Hopf boundary lemma, 16, 178

boundary blow-up problem, 67, 83 rate, 95, 105, 111

Keller-Osserman result, 84 Kelvin transformation, 146 Krein-Rutman Theorem, 2

condition exterior cone, 177 interior sphere, 177

Liouville type theorem, 140, 146 over half space, 152 over the entire space, 146, 154, 161

cone, 1 reproducing, 1 solid, 1 total, 1

locally quasi-monotone, 118 logistic equation, 61 classical, 61 degenerate, 64 over RN, 128 with weights, 102

imbedding theorems, 169, 170

convex function trick, 114, 115 De Girogi conjecture, 139, 145

maximum principle, 9, 177, 178 Aleksandrov weak, 179 Bony's, 15 property, 9 strong, 9, 28, 177, 178 weak, 177, 178

eigenvalue, 2 principal, 6, 12 problem, 6, 9, 173 simple, 2 estimate boundary, 174 global, 175 interior, 173

method sliding, 21 moving plane, 17 super and sub-solution, 33 upper and lower solution, 33

Harnack inequality, 179 189

190

Maximum

Principles

operator boundary, 163 Dirichlet boundary, 163 Neumann boundary, 164 Robin boundary, 164 strongly uniformly elliptic, 163 uniformly elliptic, 177 regularity, 173 boundary, 174 interior, 173, 174 Safonov iteration technique, 115, 139 solution lower, 34, 54 maximal, 34, 35, 38, 59, 90, 93 maximal weak, 51 minimal, 34, 35, 38, 59, 90, 93 minimal weak, 51 profile of, 75 sub-, 33 strong, 174 super, 10 upper, 34, 54

and

Applications

weak, 171 weak lower, 40 weak upper, 40 space Holder, 164 Sobolev, 166 Star of David, 21 symmetry in a half space, 117 over bounded domain, 17 over the entire space, 23, 139, 146 partial, 139 radial, 20 Steiner, 18, 23 theory L p , 174 Schauder, 163 trace, 172 weak sweeping principle, 120

Series on Partial Differential Equations and Applications- Vol. 2

Order Structure and Topological Methods in Nonlinear Partial Differential Equations The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems. The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time.

World Scientific www.worldscientific.com 5999 he

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