The Minimal Period Problem for Nonconvex Even Second Order Hamiltonian Systems

Transcription

1 Ž JOURAL OF MAEMAICAL AALYSIS AD APPLICAIOS 5, ARICLE O AY he Minimal Period Problem for onconvex Even Second Order amiltonian Systems Guihua Fei and ixiang Wang Department of Mathematics, Uniersity of Connecticut, Storrs, Connecticut 669 Submitted by Jean Mawhin Received January, 997 In this paper, we study the minimal period problem for even autonomous second order amiltonian systems defined on without any convexity assumption By using the variational methods, we obtain estimates on the minimal period of the corresponding nonconstant periodic solution of the superquadratic and asymptotically linear amiltonian systems 997 Academic Press IRODUCIO AD MAI RESULS In his pioneering work, P Rabinowitz studied the following classical amiltonian systems x VŽ x, x, Ž where is a positive integer V: is a function and V denotes its gradient In the text of this paper, we denote by a b and a the usual inner product and norm in, respectively Under the conditions that the potential function V is non-negative and superquadratic at both the origin and the infinity, ie, there exist constants and r such that VŽ x VŽ x x, xr, Ž Rabinowitz proved that the system Ž possesses a nonconstant periodic solution with any prescribed period Moreover, Rabinowitz conjectured that Ž possesses a nonconstant solution with any prescribed minimal period under his conditions Since then, there are many papers on this minimal period problem Žcf 7, 9, 8, Among these results, most of them assumed that V satisfies some kinds of convexity conditions X97 $5 Copyright 997 by Academic Press All rights of reproduction in any form reserved

2 544 FEI AD WAG Žcf 6, 7 here are only a few papers dealing with the nonconvex case Žcf, Recently, in his papers 57, Long generalized some ideas of Ekeland and ofer Žcf 6, 7 to the second order amiltonian systems without any convexity assumptions In 5, by using the natural -symmetry possessed by the system Ž and a Morse index theory method, Long obtained results on the minimal period problem for the system Ž In 6, Long studied the system Ž when V is even In this case, the system Ž possesses a natural 4-symmetry, where 4 is the Klein Fourgroup Long established estimates on the minimal period of solutions of Ž in terms of their Morse indices Ž see heorem 4 Combining this with the Mountain Pass theorem, under Rabinowitz s superquadratic condition Ž, Long proved that for every, the system Ž possesses a nonconstant -periodic solution with minimal period or 3 In this paper, we further use the ideas of 6 to study the minimal period problem of Ž We denote by L Ž s the set of all real symmetric matrices, and L Ž h L Ž s s h is semi-positive definite 4 We assume the following conditions on V Ž Ž Ž V V C, and there exists h L such that s Ž Ž V x hxxv x, x Ž V Ž Ž V is even, ie, V x V x, x Ž V3 V Ž x ožx as x We ll study the minimal period problem of Ž in the case that h his is motivated by 9 In 9 Rabinowitz considered the existence of nonconstant periodic solutions of Ž in the case that h is positive definite We ll give estimates on the minimal period of the nonconstant periodic solutions of Ž in the case h For any and h L Ž, let s D h Ž k I h k, 3 kž Ž Ž where I is the identity matrix in L Ž We define the indices of h by Ý k Ý k k k s Ž Ž i Ž h M D Ž h, Ž h M D Ž h, Ž 4 Ž Ž Ž where M, M, and M, denote the negative definite, the null, and the positive definite subspace of the selfadjoint linear operator defining it, respectively

3 MIIMAL PERIOD PROBLEM 545 First we consider the superquadratic amiltonian systems, ie, V satisfies Ž Ž Ž V4 V x V, x Ž V5 here exist constants, r,, and d such that V Ž x V Ž x xd x, xr Ž V6 V Ž x x,as x We have the following result: Ž Ž Ž EOREM Suppose V satisfies V V6 with h L s hen for eery, the system Ž possesses a nonconstant -periodic solution with minimal period k for some odd integer k satisfying Ž k i Ž h Ž h 3 By a straightforward computation, we know that heorem is a generalization of the corresponding results in 6, heorem and heorem Ž see Corollary 3 and Remark 33 ext we consider the asymptotically linear amiltonian systems, ie, V satisfies Ž V7 here exists h L Ž such that Ž Ž Ž s G x V x h xxo x, as x In the following, we always denote by w and w the largest eigenvalue of h and h, respectively EOREM Suppose V satisfies Ž V Ž V3, Ž V7, and w w hen for eery Ž ' w, w, the system Ž ' possesses a nonconstant periodic solution with minimal period his is motivated by 7 In 7, heorem Long proved a similar Ž Ž result for V C, In the case V C,, our heorem is a strict generalization of 7, heorem See Remark 4 for the comparison For the Landesman Lazer type conditions, ie, Ž Ž Ž V8 G x is bounded and G x as x, we have Ž Ž Ž EOREM 3 Suppose V satisfies V V3 and V7 Ž Ž i Assume V8 holds hen for eery satisfying i h h i h,i h h, 5 Ž Ž Ž Ž Ž Ž

4 546 FEI AD WAG Ž the system possesses a nonconstant -periodic solution with minimal period k for some odd integer k satisfying Ž k i Ž h Ž h Ž Ž ii Assume V8 holds hen for eery satisfying iž h iž h,iž h Ž h, Ž 6 Ž the system possesses a nonconstant -periodic solution with minimal period k for some odd integer k satisfying k i Ž h Ž Ž Ž Ž EOREM 4 Suppose V satisfies V V4, V7, V8, and Ž Ž V9 h, h, h h L and h h h h s hen for eery satisfying iž h Ž h iž h Ž h, Ž 7 the conclusion of heorem holds As a consequence, we have COROLLARY 5 Under the conditions of heorem 4, if w w, then for eery ' w, w,the system Ž ' possesses a nonconstant periodic solution with minimal period or 3 Remark 6 he conditions Ž 5 Ž 7 can be satisfied by many matrices in L Ž Ž see Corollary 6 s Corollary 5 may be regarded as a complement of heorem he paper is organized as follows In section, we describe the 4- symmetry and compute the symmetric Morse indices In Section 3, we use the saddle point theorem to prove heorem In Section 4, we consider the asymptotically linear cases and prove the remaining theorems E SYMMERIC MORSE IDICES AD COMPUAIO Ž, For, let S and E W ŽS, with the usual norm ž / Ž x x x dt, xe

5 MIIMAL PERIOD PROBLEM 547 We define the mentioned 4-action for any -periodic measurable func- tion x: S with,,, 4 by 4 3 xx, xž t xž t, ž / ž / xž t x t, 3xŽ t x t, ae hey are commutative and satisfying id and 3 he group is isoporphic to the Klein fourgroup 3 4 DEFIIIO For, a -periodic measurable function x: S is symmetric, if it satisfies x x, 4 ote that a -periodic function is symmetric if and only if it is even about t and, and is odd about t 4 and 34 Define SE z E z z, 44 hen SE is a closed subspace of E On SE, the norm x is equivalent to the norm ž Ž / x x t dt Recall that SE consists of those z L ŽS, whose Fourier series satisfies zž t a cos Ž k t, a, Ž he inner product in SE For z E, we define Ý k k k ž / ž / Ý Ž k k z k a is given by ž / Ý k k k ² z, z: Ž k a a Ž fž z z VŽ z dt Ž 3

6 548 FEI AD WAG ote that f is -invariant if V is even Žcf 6 4 he following proposition is given by Long Ž cf 6 Ž PROPOSIIO Suppose V C, is een hen for eery we hae Ž Ž fc SE,, and there hold ² Ž : Ž f x, y xyv x y dt, x, yse, ² Ž : Ž f x y, z yzv x yz dt, x, y, zse Ž If x SE is a critical point of f on SE, then x is a symmetric C 3 ŽS, -solution of Ž Ž 3 Conersely, if x C 3 ŽS, is a solution of Ž, and is symmetric, then x SE and x is a critical point of f on SE Ž For given, we suppose that At Define satisfies the condition Ž AS A CS Ž, L Ž, and it is even about t and 4 ² : s Ž A z, z AŽ t zz dt, z, z SE, Ž 4 hen A : SE SE is a linear compact selfadjoint operator on SE Ž cf 6 Moreover, we have Set dim M Ž id A, dim M Ž id A Ž 5 Ž y, z y z AŽ t yz dt, y, zse Ž hen by 4 we have Ž ² Ž : Ž y, z id A y, z, y, zse 6 DEFIIIO 3 Define Ž Ž si AŽ t dim M Ž id A, s AŽ t dim M Ž id A si Ž AŽ t and s Ž At Ž are called the symmetric Morse indices of At Ž

7 MIIMAL PERIOD PROBLEM 549 For every nonconstant C 3 ŽS, -solution x of Ž which is even about t and odd about t 4, let At Ž VŽxt Ž If V is even, then At Ž satisfies the condition Ž AS So the symmetric Morse indices of x, denoted by si Ž x and s Ž x, can be defined as si Ž x si Ž AŽ t, s Ž x s Ž AŽ t By Proposition, x is a critical point of f on SE, and f Ž x defines the bilinear form Ž 6 on SE hus si Ž x and s Ž x are just the Morse index and nullity of f at x in SE Let OŽ x be the greatest positive integer k such that x is k-periodic In 6, Long proved the following theorem which estimates OŽ x in term of si Ž x Ž EOREM 4 Suppose V C, is een For and eery nonconstant C 3 ŽS, -solution x of Ž which is een about t and odd about t 4, there holds Ž OŽ x si Ž x ote that for any and any h L Ž, At Ž s hsatisfies the condition Ž AS In this case, we can compute the symmetric Morse indices of h directly Ž EOREM 5 For any and h L s, there holds siž h iž h, sž h Ž h Proof Let A be the operator defined by Ž 4 on SE corresponding to h By Ž and Ž, the operator A has explicit expressions ž / A zý hak cosž Ž k t, Ž k k where zt Ž SE with the form Ž hus Ý k ž ž Ž k / / ž k / / Ž id A z I h a cos Ž k t Ž By 3 and a straightforward computation, we have Ž 7 dim M id A Ý dim M Ž Dk h,, 8 k Ž Ž Ž Ž Combining this with 3 and Definition 3 yields the conclusions

8 55 FEI AD WAG Ž Ž Ž COROLLARY 6 i For h, h L, if h h L, then s s siž h s Ž h siž h s Ž h, Ž Ž ii If h L, for any, we hae s siž h s Ž h siž h s Ž h Ž iii Assume h L Ž s and w is the largest eigenalue of h Ž a If w, for eery, there holds si Ž h s Ž h Ž 9 Ž b If w, for eery w, Ž 9 holds For eery ' w, there holds ' Moreoer, if w, we hae ' si Ž h s Ž h siž h Proof By Ž 3, Ž 7, Ž 8, and a direct computation, we obtain Ž i and Ž ii ow we prove Ž iii Let w be the eigenvalues of h By Ž 3 and heorem 5 it is easy to show that ½ ž / m 5 si Ž h Ž k, m Ž k,m,k,,, Ž ½ ž / m 5 s Ž h Ž k, m Ž k,m,k,, ' Ž By Ž and Ž, we get Ž a If w and w, then wž hus Ž 9 holds For ' w, by Ž and Ž we have si Ž h s Ž h Ž, 4 Similarly, ' w, we have si Ž h Ž, 4 he proof is complete

9 MIIMAL PERIOD PROBLEM 55 3 SUPERQUADRAIC AMILOIA SYSEMS In this section, we study the minimal period problem for the superquadratic amiltonian systems Ž For, let E, SE, and 4 be defined as in Section, and f be defined by Ž 3 hen by Proposition we know that looking for the symmetric C 3 ŽS, -solutions of Ž is equivalent to looking for critical points of f on SE In order to find the critical points of f, we need the following saddle point theorem which was proved in 8, 4, EOREM 3 Let E be the ilbert space with orthogonal decomposition EXY, where dim X Suppose f C Ž E,, satisfies the Ž PS condition, and the following conditions: Ž F here exist and such that fž w, w B Ž Y Ž F here exist e B Ž Y and r such that fž w, w Q, where Q ŽB Ž X re r r 4 hen Ž r f possesses a critical alue c, which is gien by Ž c inf max f hž w, h wq where h CQ,E Ž hid on Q 4 Ž here exists an element w K w E fž w, fž w c4 c such that the negatie Morse index iž w of f at w satisfies iž w dim X Proof of heorem For, let A be the operator defined by Ž 4 on SE corresponding to h By Ž V, Ž 3, and Ž 6 we have ² : fž z Ž id A z, z VŽ x dt, zse Ž 3 We carry out the proof in several steps Step Let X M Ž id A M Ž id A, Y M Ž id A hen the Sobolev inequality Žcf 8 implies that / y max yž t y ž, yy Ž 3 t, Ž Combining this with V3 yields that, for y Y, Ž ² Ž : Ž f y id A y, y o y as y

10 55 FEI AD WAG his implies that there exists, such that fž y, ybž Y Ž 33 Let e B Ž Y and set Q re r r4 Br Ž X 4, where r is free for the moment By Ž V6 there exist constants c id A e L 4dŽ, c Ž 34 such that VŽ x c x c, x ; Ž 35 For z z z X, by Ž 3, Ž 34, Ž 35, and Ž 5 we have ² : ² : Ž fž rez Ž id A z, z r Ž id A e, e VŽ zre dt r id A id A z 3 4 c zre dt c id A r c z c z c hen there exists r such that fž z, z Q Ž 36 Step f satisfies the Ž PS condition on SE, ie, any sequence u SE satisfying fž u k k M and fž uk as k Ž 37 possesses a subsequence convergent in SE In fact, for large k and u u, by Ž 3, Ž 34, Ž 35, Ž 37, and Ž V5 k we have Mu fž u ² fž u,u: VŽ u uv Ž u dt ž / d V Ž u dt u dtm V Ž u dtm 4 3 L 4 M u M

11 MIIMAL PERIOD PROBLEM 553 his implies that L 5 6 u M Ž u, VŽ u dtm Ž u Ž 38 Ž Ž Ž ow by 3, 37, and 38 we have Ž ² : Ž u f u A u,u V u dt MM7Ž u his implies that u 4 k is bounded A standard argument shows that f satisfies the Ž PS condition Step 3 By heorem 3, there exists a critical point x SE of f with fž x c and the Morse index m Ž x of f at x on SE satisfies m Ž x dim M Ž id A dim M Ž id A Ž 39 By Ž V4 and Proposition, x is a nonconstant symmetric C 3 ŽS, - solution of Ž hus by Definition 3, heorem 4, heorem 5, and Ž 39 we have Ž Ž OŽ x si Ž x m Ž x i Ž h Ž h 3 Ž By Lemma of 6, O x is odd he proof is complete Let w be the largest eigenvalue of h hen by heorem 5 and Corollary 6, for every w, we have ' siž h s Ž h hus we have the following corollary Ž Ž Ž COROLLARY 3 Suppose that V satisfies V V6 with h L s and the greatest eigenalue of h is w hen for eery ' w, the system Ž possesses a nonconstant -periodic solution with minimal period or3, and which is een about t and, and odd about t 4 and 34 Remark 33 Ž i It is easy to show that the conditions Ž V5 and Ž V6 are equivalent to the usual superquadratic condition Ž For example, if V Ž x satisfies Ž, then V Ž x satisfies Ž V5 and Ž V6 with d ere we use the conditions Ž V5 and Ž V6 because sometime they are easy for applications

12 554 FEI AD WAG Ž ii If h, by Corollary 6 we have si Ž h s Ž h for any In this case we get back the result due to Long Ž cf 6, heorem Ž iii A similar result as Corollary 3 was proved by Long in 6 But they required that ' w So Corollary 3 extends heorem in 6 4 ASYMPOICALLY LIEAR AMILOIA SYSEMS In this section, we study the minimal period problem for the system Ž under the asymptotically linear conditions For, let E and SE be defined as in Section, and f be defined by Ž 3 Let A and A be the operator, defined by Ž 4 on SE, corresponding to h and h, respectively We first prove heorem ' Proof of heorem Since w, by Corollary 6 we have that Ž 9 holds for h hus id A is positive definite in SE On the other hand, since w, Corollary 6 implies that ' siž h Ž 4 ote that ut Ž dt for any u SE hen by Wirtinger s inequality we have Ž u Ž 4 u, use Ž 4 L Ž Ž By V7, there exist c id A and c such that Ž Ž G x c x c, x 43 Ž Ž Ž Ž ow by 3, V7, 4, and 43, for z SE, we have Ž ² Ž : Ž f z id A z, z G z dt Ž Ž Ž id A z c 4 z c Ž id A z c 4 herefore f is coercive on SE, ie, fž z, as z

13 MIIMAL PERIOD PROBLEM 555 ote that f is weal lower semicontinuous on SE Žcf 7, 8 hen the functional f attains its minimum on SE at some point y SE Žcf 8, heorem y is a critical point of f and the Morse index si Ž y of f at y satisfies siž y Ž 44 hus by Ž 4 and Proposition, y and y is a nonconstant symmetric C 3 ŽS, -solution of Ž ow Ž 44 and heorem 4 implies that he proof is complete Ž OŽ y si Ž y Remark 4 In 6, heorem, under the assumptions that V Ž C, satisfies Ž V and the following m VŽ x x b, x, Ž 45 M VŽ x x, x, Ž 46 where b,, and M m, Long proved that for every Ž ' M,' m, the system Ž possesses a nonconstant periodic solution with minimal period ote that in heorem if Ž V7 is replaced by Ž 45, the conclusion also holds with w m But if we apply 6, heorem to our case, Ž 46 implies that the smallest eigenvalue ws of h should satisfy ws w, ie, h is positive definite ere in heorem, we do not need that h is positive definite In fact, h may have Ž negative eigenvalues herefore, if V C,, our heorem is a strict generalization of 6, heorem In order to prove heorem 3, we need the following definition and theorem introduced and proved in 8 DEFIIIO 4 8 Let E be a C -Riemannian manifold, D a closed subset of E A family FŽ is said to be a homological family of dimension q with boundary D if for some nontrivial class Ž E, D q the family FŽ is defined by 4 FŽ G E: is in the image of i: Ž G, D Ž E, D, where i is the homomorphism induced by the immersion i: G E EOREM 43 8 As in Definition 4, for gien E, D, and, let FŽ be a homological family of dimension q with boundary D Suppose that fc Ž E, R satisfies the Ž PS condition Define c cž f, FŽ inf sup fž w Ž 47 q GFŽ wg q

14 556 FEI AD WAG Suppose that sup w D fž w c and f is Fredholm on 4 K x E: fž x, fž x c Ž 48 c hen there exists x K such that the Morse indices m Ž x and m Ž x c of the functional f at x satisfy Proof of heorem 3 q m Ž x m Ž x q For satisfying Ž 5, set X M Ž id A M Ž id A, Y M Ž id A For z Y, by Ž 3, Ž V7, and ŽV8, we have Ž ² Ž : Ž f z id A z, z G z dt Ž id A z c z Ž id A c Ž 49 For z z z X, by Ž V7 and ŽV8, we have Ž ² Ž : Ž f z id A z, z G z dt id A z c z G z dt 4 Ž Ž Ž Ž Ž Since dim M id A, by V8 we have GŽ z dt, as z Ž 4 Combining this with Ž 4 yields that there exists r and such that fž z, zq, Ž 4 where Q z X z r 4 It is well known that, under the conditions Ž V7 and ŽV8, f satisfies the Ž PS condition Žcf 5, 3 Let S Y hen Q and S are homologically linked Žcf 5, 8 Let DQ and Q Ž SE, D k, where k dim X hen is non- trivial and FŽ defined by Definition 4 is a homological family of dimension k with boundary D Žcf 5, 8 It is well known that f is Fredholm on K defined by Ž 47 and Ž 48 By Ž 49 and Ž 4 we obtain c Ž sup fž z cc f, FŽ zd

15 MIIMAL PERIOD PROBLEM 557 Žcf 5 hus by heorem 43, there exists x Kc indices m Ž x and m Ž x of f at x satisfy such that the Morse dim X m Ž x m Ž x dim X Combining this with Proposition, Ž 5, and heorem 5 yields that x and x is a nonconstant symmetric C ŽS, -solution of Ž which satisfies siž x iž h Ž h Ž Ž hus by heorem 4 we get the conclusion of i he proof of ii is similar We omit the details Proof of heorem 4 For satisfying Ž 7, let X M Ž id A M Ž id A, Y M Ž id A Using the same arguments as Step in the proof of heorem, we get Ž Ž 33 By 7 and heorem 5 we have Ž Z M Ž id A M Ž id A M Ž id A 4 Ž Let e B Z and set By Ž V9 we have that A A 4 Q re:rr 4 B Ž X r is semi-positive definite and Ž id AŽ id A Ž id AŽ id A ence for any z X, we have ² Ž : ² Ž : ² Ž : A A e, z id A e, z id A e, z hus for any z re z z Q, by Ž V7 and ŽV8 we obtain Ž ² Ž : Ž f z id A z, z G z dt r id A e, e id A z, z ² Ž : ² Ž : ² Ž Ž : Ž A A z z, z z G z dt Ž id A z M z GŽ zre dt

16 558 FEI AD WAG ow by Ž 5 and ŽV8, there exists r such that Ž 36 holds By standard arguments, Ž V7 and ŽV8 imply that f satisfies the Ž PS condition Žcf 5, 8, 3 Using the same arguments as Step 3 in the proof of heorem, we get the conclusion he proof is complete Proof of Corollary 5 For any w, w, by heorem 5 and Corollary 6, we have iž h Ž h iž h Ž h he conclusion follows from heorem 4 ' ' ACKOWLEDGMES he authors express their sincere thanks to Professor Yiming Long and Professor Wenzhao Lu for their help and useful suggestions REFERECES Amann and E Zehnder, ontrival solutions for a class of nonresonance problem and applications to nonlinear differential equations, Ann Scuola orm Sup Pisa Cl Sci () 4 7 Ž 98, A Ambrosetti and V Coti Zelati, Solutions with minimal period for amiltonian systems in a potential well, Ann Inst Poincare Anal on Lineaire 4 Ž 987, A Ambrosetti and G Mancini, Solutions of minimal period for a class of convex amiltonian systems, Math Ann 55 Ž 98, F Clarke and I Ekeland, amiltonian trajectories having prescribed minimal period, Comm Pure Appl Math 33 Ž 98, 36 5 K C Chang, Infinite dimensional Morse theory and multiple solution problems, in Progress in onlinear Differential Equations and heir Applciations, Vol 6, Birkhauser, Basel, I Ekeland and ofer, Periodic solution with prescribed period for convex autonomous amiltonian systems, Inent Math 8 Ž 985, I Ekeland, Convexity Method in amiltonian Mechanics, Springer-Verlag, Berlin, 99 8 Ghoussoub, Location, multiplicity and Morse indices of min-max critical points, J Reine Angew Math 47 Ž 99, M Giradi and M Matzeu, Some results on solution of minimal period to superquadratic amiltonian equations, onlinear Anal 7 Ž 983, M Giradi and M Matzeu, Solution of minimal period for a class of nonconvex amiltonian systems and applications to the fixed energy problem, onlinear Anal Ž 986, 3738 M Giradi and M Matzeu, Dual Morse index estimates for periodic solutions of amiltonian systems in some nonconvex superquadratic case, onlinear Anal 7 Ž 99, M Giradi and M Matzeu, Essential critical points of linking type and solution of minimal period to superquadratic amiltonian systems, preprint, 99

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