Periodic solutions of nonlinear wave equations with non-monotone forcing terms

Periodi solutions of nonlinear wave equations with non-monotone for ing terms Massimiliano Berti, Lu a Bias o



Abstra t Existen e and regularity of periodi solutions of nonlinear, ompletely resonant, for ed wave equations is proved for a large lass of non-monotone for ing terms. Our approa h is based on a variational Lyapunov-S hmidt redu tion. The orresponding in nite dimensional bifur ation equation exhibits an intrinsi la k of ompa tness. This diÆ ulty is over ome nding a-priori estimates for the onstrained minimizers of the redu ed a tion fun tional, through te hniques inspired by regularity theory as in [R67℄.

Abstra t

Presentiamo risultati di esistenza ed uni ita di soluzioni periodi he per equazioni delle onde nonlineari,

ompletamente risonanti e periodi amente forzate nel tempo, per un'ampia lasse di termini forzanti non monotoni. Il nostro appro

io si basa su una riduzione variazionale di tipo Lyapunov-S hmidt. La orrispondente equazione di bifor azione man a radi almente di proprieta di ompattezza. Questa diÆ olta viene superata trovando opportune stime a-priori per i minimi vin olati del funzionale di azione ridotto, mediante te ni he ispirate alla teoria della regolarita di [R67℄. 1 Key words:

Wave Equation, Periodi Solutions, Variational Methods, A-priori Estimates, Lyapunov-S hmidt

Redu tion.

MSC lassi ation: 1

35L05, 35L20, 35B10, 37K10.

Introdu tion and Results

We outline in this Note the re ent results obtained in [BBi04℄ on erning existen e and regularity of nontrivial time-periodi solutions for ompletely resonant, nonlinear, for ed wave equations like u = "f (t; x; u; ") (1.1) with Diri hlet boundary onditions (1.2) u(t; 0) = u(t;  ) = 0 where  := tt xx is the D'Alembertian operator, " is a small parameter and the nonlinear for ing term f (t; x; u; ") is T -periodi in time. We onsider the ase when T is a rational multiple of 2 and, for simpli ity of exposition, we assume T = 2 : We look for 2-periodi in time solutions of (1.1)-(1.2), namely u satisfying u(t + 2; x) = u(t; x) : (1.3)  Massimiliano Berti, SISSA, via Beirut 2-4, Trieste, Italy, bertisissa.it . Lu a Bias o, Universit a di Roma 3, Largo S. Leonardo Murialdo, Roma, Italy, bias omat.uniroma3.it . 1 Supported by MIUR Variational Methods and Nonlinear Dierential Equations.

1

2

M. Berti, L. Bias o

For " = 0, (1.1)-(1.2) redu es to the linear homogeneous wave equation (

u = 0 (1.4) u(t; 0) = u(t;  ) = 0 whi h possesses an in nite dimensional spa e of solutions whi h are 2-periodi in time and of the form v (t; x) = v^(t + x) v^(t x) for any 2 -periodi fun tion v^(
). For this reason equation (1.1)-(1.2) is alled

ompletely resonant. 6 0 is to nd from whi h The main diÆ ulty for proving existen e of solutions of (1.1)-(1.2)-(1.3) for " = periodi orbits of the linear equation (1.4) the solutions of the nonlinear equation (1.1) bran h o. This requires to solve an in nite dimensional bifur ation equation with an intrinsi la k of ompa tness. The rst breakthrough regarding problem (1.1)-(1.2)-(1.3) was a hieved by Rabinowitz in [R67℄ where existen e and regularity of solutions was proved for nonlinearities satisfying the strongly monotone assumption (u f )(t; x; u)  > 0. Using methods inspired by the theory of ellipti regularity, [R67℄ proved the existen e of a unique urve of smooth solutions for " small. Other existen e results have been obtained, still for strongly monotone f 's, in [DST68℄-[BN78℄. Subsequently, Rabinowitz [R71℄ was able to prove existen e of weak solutions of (1.1)-(1.2)-(1.3) for weakly monotone nonlinearities like f (t; x; u) = u2k+1 + G(t; x; u) where G(t; x; u2 )  G(t; x; u1 ) if u2  u1 . A tually, in [R71℄ the bifur ation of a global ontinuum bran h of weak solutions is proved. In all the quoted papers the monotoni ity assumption (strong or weak) is the key property for over oming the la k of ompa tness in the in nite dimensional bifur ation equation. We underline that, in general, the weak solutions obtained in [R71℄ are only ontinuous fun tions. Con erning regularity, Brezis and Nirenberg [BN78℄ proved -but only for strongly monotone nonlinearitiesthat any L1 -solution of (1.1)-(1.2)-(1.3) is smooth, even in the nonperturbative ase " = 1, whenever f is smooth. On the other hand, very little is known about existen e and regularity of solutions if we drop the monotoni ity assumption on the for ing term f . Willem [W81℄, Hofer [H82℄ and Coron [C83℄ have onsidered the lass of equations (1.1)-(1.2) where f (t; x; u) = g(u)+ h(t; x), " = 1, and g(u) satis es suitable linear growth onditions. Existen e of weak solutions is proved, in [W81℄-[H82℄, for a set of h dense in L2 , although expli it riteria that hara terize su h h are not provided. The in nite dimensional bifur ation problem is over ome by assuming non-resonan e hypothesys between the asymptoti behaviour of g(u) and the spe trum of . On the other side, Coron [C83℄ nds weak solutions assuming the additional symmetry h(t; x) = h(t + ;  x) and restri ting to the spa e of fun tions satisfying u(t; x) = u(t + ;  x), where the Kernel of the d'Alembertian operator  redu es to 0. Let us now present the results obtained in [BBi04℄ on existen e and regularity of solutions of (1.1)(1.2)-(1.3) for a large lass of nonmonotone for ing terms f (t; x; u). We look for solutions u : ! R of (1.1)-(1.2)-(1.3) in the Bana h spa e E := H 1 ( ) \ C01=2 ( );

:= T  (0; )

where H 1 ( ) is the usual Sobolev spa e and C01=2 ( ) is the spa e of all the 1=2-Holder ontinuous fun tions u : ! R satisfying (1.2), endowed with norm kukE := kukH 1 ( ) + kukC 1=2( ) where kuk2H 1( ) := kuk2L2( ) + kuxk2L2 ( ) + kut k2L2( ) and

kukC 1=2( ) := kukC 0( ) +

ju(t; x) u(t1 ; x1 )j ( j t t1 j + jx x1 j)1=2 (t;x)6=(t1 ;x1 ) sup

:

3

Periodi solutions with nonmonotone for ing terms

Criti al points of the Lagrangian a tion fun tional 2 C 1 (E; R) (u) :=

Z h 2 ut

u2x

i

(1.5) 2 + "F (t; x; u; ") dtdx ;

2 R where F (t; x; u; ") := 0u f (t; x;  ; ")d , are weak solutions of (1.1)-(1.2)-(1.3). For " = 0, the riti al points of in E redu e to the solutions of the linear equation (1.4) and form the subspa e V := N \ H 1 ( ) where N

:=



v (t; x) = v^(t + x)

v^(t



x) v^ 2 L2 (T) and

Z 2

0

v^(s) ds = 0



:

(1.6)

Note that V =fv(t; x) = v^(t + x) v^(t x) 2 N j v^ 2 H 1 (T)g  E sin e any fun tion v^ 2 H 1 (T) is 1=2-Holder ontinuous. R Let N ? := fh 2 L2 ( ) j hv = 0; 8v 2 N g denote the L2-orthogonal of N whi h oin ides with the range of  in L2( ). In [BBi04℄ we prove the following Theorem: Theorem 1 Let f (t; x; u) = u2k + h(t; x) and h 2 N ? satis es h(t; x) > 0 (or h(t; x) < 0) a.e. in . Then, for " small enough, there exists at least one weak solution u 2 E of (1.1)-(1.2)-(1.3) with kukE  C j"j. If, moreover, h 2 H j ( ) \ C j 1 ( ), j  1, then u 2 H j +1 ( ) \ C0j ( ) with kukH j+1 ( ) + kukC j ( )  C j"j and therefore, for j  2; u is a lassi al solution. Theorem 1 is a Corollary of the following more general result whi h enables to deal with non-monotone nonlinearities like, for example, f (t; x; u) = (sin x) u2k + h(t; x), f (t; x; u) =  u2k + u2k+1 + h(t; x). Theorem 2 Let f (t; x; u) = g (t; x; u) + h(t; x), h(t; x) 2 N ? and g (t; x; u) := (x)u2k + R(t; x; u) where R; t R, u R 2 C (  R; R ) satisfy2 kR(
; u)kC ( ) = o(u2k ) ; ktR(
; u)kC ( ) = O(u2k ) ; ku R(
; u)kC ( ) = o(u2k 1 ) ; (1.7) and 2 C ([0;  ℄; R) veri es, for x 2 (0;  ), (x) > 0 (or (x) < 0) and ( x) = (x). (i) (Existen e) Assume there exists a weak solution H 2 E of H = h su h that

8(t; x) 2 : (1.8) Then, for " small enough, there exists at least one weak solution u 2 E of (1.1)-(1.2)-(1.3) satisfying kukE  C j"j.
(ii) (Regularity) If, moreover, h 2 H j ( ) \ C j 1 ( ), 2 H j (0; ) ; R; t R, u R 2 C j (  R), j  1, then u 2 H j +1 ( ) \ C0j ( ) and, for j  2, u is a lassi al solution. Note that Theorem 2 does not require any growth ondition on g at in nity. In parti ular it applies for any analyti fun tion g(u) satisfying g(0) = g0 (0) = : : : = g2k 1 (0) = 0 and g2k (0) 6= 0. We now olle t some omments on the previous results. Remark 1.1 The assumption h 2 N ? is not of te hni al nature both in Theorem 1 and in Theorem 2 (at least if g = g (x; u) = g (x; u) = g ( x; u)). A tually one an prove that, if h 2= N ? , periodi solutions of problem (1.1)-(1.2)-(1.3) do not exist in any xed ball fkukL1  Rg; R > 0, for " small. H (t; x) > 0

(or H (t; x) < 0)

2 The notation f (z ) = o(z p ), p 2 N, means that f (z )=jzjp ! 0 as z ! 0. f (z ) = O (z p ) means that there exists a onstant C > 0 su h that jf (z )j  Cjzjp for all z in a neighboorhood of 0.

4

M. Berti, L. Bias o

In Theorem 2 hypothesys (1.8) and > 0 (or < 0) are assumed to prove the existen e of a minimum of the \redu ed a tion fun tional" , see (1.17). A suÆ ient ondition implying (1.8) is h > 0 a.e. in . This follows by the \maximum prin iple" proposition Remark 1.2

h 2 N ? ; h > 0 a.e. in

=) 9 H 2 E

solving

H = h

with H > 0 :

(1.9)

This is the key step to derive Theorem 1 from Theorem 2. It is not at all obvious that the weak solution u of Theorems 1, 2 is a tually smooth. Indeed, while regularity always holds true for stri tly monotone nonlinearities (see [R67℄-[BN78℄), yet for weakly monotone f it is not proved in general, unless the weak solution u veri es kN ukL2  C > 0 (see [R71℄). Note, on the ontrary, that the weak solution u of Theorem 2 satis es kN ukL2 = O("). Moreover, assuming

Remark 1.3 (Regularity)

ktl xm un RkC ( ) = O(u2k n ) ; 8 0  l; n  j + 1 ; 0  m  j ; l + m + n  j + 1 we an also prove the estimate

kukH j+1 ( ) + kukC j ( )  C j"j :

(1.10) (1.11)

Remark 1.4 (Multipli ity) For nonmonotone nonlinearities f one an not in general expe t uni ity of the solutions. A tually, for f (t; x; u) = g (x; u) + h(t; x) with g (x; u) = g (x; u), g ( x; u) = g (x; u), there exist in nitely many h 2 N ? for whi h problem (1.1)-(1.2)-(1.3) has (at least) 3 solutions. Remark 1.5 (Mimimal period)

u(t; x) has minimal period 2.

If h(t; x) has minimal period 2 w.r.t time, then also the solution

Finally, we extend the result of [R67℄ proving existen e of periodi solutions for nonmonotone nonlinearities f (t; x; u) obtained adding to a nonlinearity fe(t; x; u) as in [R67℄ (i.e. u fe  > 0) any nonmonotone term a(x; u) satisfying a(x; u) = a(x; u) ; a( x; u) = a(x; u) (1.12) or a(x; u) = a(x; u) ; a( x; u) = a(x; u) : (1.13) A prototype nonlinearity is f (t; x; u) = u2k + fe(t; x; u) with u fe  > 0. Theorem 3 Let f (t; x; u) = fe(t; x; u)+ a(x; u) where f, t f, u f are ontinuous, u fe  > 0 and a(x; u) satisfy (1.12) or (1.13). Then, for " small enough, (1.1)-(1.2)-(1.3) has at least one weak solution u 2 E. If moreover f, t f, u f 2 C j (  R), j  1, then u 2 H j +1 ( ) \ C0j ( ). In the next subse tion we des ribe the method followed in [BBi04℄ to prove Theorems 1, 2, 3. We anti ipate that su h approa h is not merely a sharpening of the ideas of [R67℄-[R71℄ whi h, to deal with non monotone nonlinearities, require a signi ant hange of prospe tive. 1.1

Sket h of the Proof

We look for riti al points of the Lagrangian a tion fun tional : E ! R de ned in (1.5) performing a variational Lyapunov-S hmidt redu tion. We de ompose the spa e E as E =V W where V := N \ H 1 ( ) and W := N ? \ H 1 ( ) \ C01=2 ( ) :

Periodi solutions with nonmonotone for ing terms

5

Setting u = v + w with v 2 V , w 2 W and denoting by N and N ? the proje tors from L2 ( ) onto N and N ? respe tively, problem (1.1)-(1.2)-(1.3) is equivalent to solve the bifur ation equation N f (v + w; ") = 0 (1.14) and the range equation w = " 1 N ? f (v + w; ") (1.15) 1 ? ? where  : N ! N is the inverse of  and f (u; ") denotes the Nemitski operator asso iated to f , namely [f (u; ")℄(t; x) := f (t; x; u; "): The usual approa h of [R67℄-[DST68℄-[R71℄ is to nd, rst, by the monotoni ity of f , the unique solution v = v(w) of the bifur ation equation (1.14) and, next, to solve the range equation (1.15). On the other hand, for nonmonotone for ing terms, one an not in general solve uniquely the equation (1.14) {re all by remark 1.4 that in general uni ity of solutions does not hold. Therefore, to deal with non monotone nonlinearities, we must solve rst the range equation and thereafter the bifur ation equation. For other appli ations of this approa h to perturbation problems in riti al point theory, see e.g. the forth oming monograph by Ambrosetti and Mal hiodi [AM04℄. We nd a solution w := w(v; ") 2 W of the range equation (1.15) satisfying kw(v; ")kE = O(") by means of a quantitative version of the Impli it Fun tion Theorem. Here no serious diÆ ulties arise sin e  1 a ting on W is a ompa t operator, due to the assumption T = 2 (a tually k 1f kE  C kf kL2 , 8f 2 L2 , see [BN78℄). Remark 1.6 More in general,  1 is ompa t on the orthogonal omplement of ker() whenever T is a rational multiple of 2. On the ontrary, if T is an irrational multiple of 2, then  1 is, in general, unbounded (a \small divisor" problem appears), but the kernel of  redu es to 0 (there is no bifur ation equation). For existen e of periodi solutions in the ase T=2 is irrational see e.g. [PY89℄. On e the range equation (1.15) has been solved by w(v; ") 2 W it remains the in nite dimensional bifur ation equation N f (v + w(v; "); ") = 0 (1.16) whi h, by the Lyapunov-S hmidt redu tion pro edure, turns out to be the Euler-Lagrange equation of the redu ed Lagrangian a tion fun tional :V !R (v) := (v + w(v; ")) : (1.17) Sin e  la ks ompa tness properties, we an not rely on riti al point theory, unlike the autonomous

ase onsidered in [BB03℄-[BB04a℄ where, thanks to the \vis ous term" kvk2H 1 , existen e and regularity of solutions is proved through the Mountain Pass Theorem and standard ellipti regularity theory. We attempt to minimize . We do not try to apply the dire t methods of the al ulus of variations be ause , even though it ould possess some oer ivity property, will not be onvex (being f non monotone). Moreover, without assuming any growth ondition on the nonlinearity f , the fun tional  ould neither be well de ned on any Lp -spa e.  We minimize  onstrained in BR := v 2 V; kvkH 1  R , 8R > 0. By standard ompa tness arguments  attains minimum at, say, v 2 BR : Sin e v ould belong to the boundary BR we an only

on lude the variational inequality Dv (v )['℄ =

Z



f (v + w(v ; "); ")'  0

for any admissible variation ' 2 V , i.e. if v + ' 2 BR , 8 < 0 suÆ iently small.

(1.18)

6

M. Berti, L. Bias o

The heart of the existen e proof of the weak solution u of Theorem 1, Theorem 2 and Theorem 3 is to obtain, hoosing suitable admissible variations the a-priori estimate kvkH 1 < R for some R > 0, i.e. to show that v is an inner minimum point of  in BR . The strong monotoni ity assumption (u f )(t; x; u)  > 0 would allow here to get su h a-priori estimates by arguments similar to [R67℄. On the ontrary, the main diÆ ulty for proving Theorems 1, 2 and 3 whi h deal with non-monotone nonlinearities is to obtain su h a priori-estimates for v. The most diÆ ult ases are the proof of Theorems 1 and 2. To understand the problem, let onsider the parti ular nonlinearity f (t; x; u) = u2k + h(t; x) of Theorem 1. The even term u2k does not give any

ontribution into the variational inequality (1.18) at the 0th-order in ", sin e the right hand side of (1.18) redu es, for " = 0, to Z   v2k + h(t; x) ' = 0; 8' 2 V

R sin e h 2 N ? and v2k '  0 by the spe i form v = v^(t + x) v^(t x), ' = '^(t + x) '^(t x) of the fun tions of V . Therefore, for deriving, if ever possible, the required a-priori estimates, we have to develop the variational inequality (1.18) at higher orders in ". We obtain 0

Z



2kv2k

1 ' w(v ; ") + O(w2 (v ; ")) =

Z



" 2k v2k 1 '  1 (h + v2k ) + O("2 )

(1.19)

be ause w(v ; ") = " 1 (v2k + h) + o(") (re all that v2k , h 2 N ? ). We now sket h how the "-order term in the variational inequality (1.19) allows to prove an L2k -estimate for v. Inserting the admissible variation ' := v in (1.19) we get Z

H v2k + v2k  1 v2k  O(")

H = h whi h veri es H (t; x) > 0

(1.20)

where H is a weak solution of in (H exists by the \maximum prin iple" proposition (1.9)). The ru ial fa t is that the rst term in (1.20) satis es the oer ivity inequality Z



Hv 2k

 (H )

Z



v 2k ;

8v 2 V

(1.21)

for some onstant (H ) > 0. We remark that (1.21) is not trivial be ause H vanishes at the boundary (H (t; 0) = H (t; ) = 0) and, indeed, its R proof relies on the spe i form v (t; x) = v^(t + x) v^(t x) of theR fun tions of V . The se ond term v2k  1v2k will be negligible, "- lose to the origin, with respe t to Hv2k and (1.20)-(1.21) will provide the L2k -estimate for v. Next, we an obtain an L1 -estimate and the required H 1 -estimate for v inserting further admissible variations ' (inspired by [R67℄) into (1.19) and using inequalities similar to (1.21). In this way we prove the existen e of a weak solution u in the interior of some BR . The regularity of the solution u -fa t not at all obvious for non-monotone nonlinearities- is proved using similar te hniques inspired to regularity theory. We insert suitable variations ' in Dv (v )['℄ = 0 and, using inequalities like (1.21), we get estimates for the L1 and H 1 -norm of the higher order derivatives of v. Theorem 2 is proved developing su h ideas and a areful analysis of the further term R. Finally, the proof of Theorem 3 is easier than for Theorems 1 and 2. Indeed the additional R term a(x; u) does not ontribute into the variational inequality (1.18) at the 0th-order in ", be ause a(x; v)'  0, 8' 2 V . Therefore the dominant term in the variational inequality (1.18) is provided by the monotone for ing term fe and the required a-priori estimates are obtained with arguments similar to [R67℄.

Periodi solutions with nonmonotone for ing terms

7

Referen es [AM04℄ Ambrosetti, A.; Mal hiodi, A. : Perturbation methods and semilinear ellipti problems on Rn , to appear. [BBi04℄ Berti, M.; Bias o, L.: For ed vibrations of wave equations with nonmonotone nonlinearities, preprint SISSA 2004. [BB03℄ Berti, M.; Bolle, P.: Periodi solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 243 (2003), no. 2, 315{328. [BB04a℄ Berti, M.; Bolle, P.: Multipli ity of periodi solutions of nonlinear wave equations, Nonlinear Anal. 56, (2004), 1011{1046. [BN78℄ Brezis, H.; Nirenberg, L. : For ed vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 1{30. [C83℄ Coron, J.-M. : Periodi solutions of a nonlinear wave equation without assumption of monotoni ity, Math. Ann. 262 (1983), no. 2, 273{285. [DST68℄ De Simon, L.; Torelli, H. : Soluzioni periodi he di equazioni a derivate parziali di tipo iperboli o non lineari, Rend. Sem. Mat. Univ. Padova 40 1968 380{401. [H82℄ Hofer, H.: On the range of a wave operator with nonmonotone nonlinearity, Math. Na hr. 106 (1982), 327{340. [PY89℄ Plotnikov, P. I.; Yungerman, L. N.: Periodi solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length, translation in Dierential Equations 24 (1988), no. 9, 1059{1065 (1989). [R67℄ Rabinowitz, P.: Periodi solutions of nonlinear hyperboli partial dierential equations, Comm. Pure Appl. Math. 20, 145{205, 1967. [R71℄ Rabinowitz, P.: Time periodi solutions of nonlinear wave equations, Manus ripta Math. 5 (1971), 165{194. [W81℄ Willem, M. Density of the range of potential operators, Pro . Amer. Math. So . 83 (1981), no. 2, 341{344.