Periodic solutions of an infinite-dimensional Hamiltonian system

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Abstract

We consider the existence of periodic solutions of the unbounded Hamiltonian system(H)tu-Δxu=Hv(t,x,u,v)-tv-Δxv=Hu(t,x,u,v)for(t,x)R×Ω.Unlike previous work, in our case the energy functional does not satisfy the Palais-Smale conditions.

Introduction

Based on a developed local linking theorem (cf. [1], [2]) we look for periodic solutions of the following supquadratic Hamiltonian system:(H)tu-xu=Hv(t,x,u,v)-tv-Δxv=Hu(t,x,u,v)for(t,x)R×Ω,where ΩRN,N1 is a bounded domain with smooth boundary Ω and H:R×Ω¯×R2R is a C1 function. SettingJ=0-110,J0=0110andz(u,v),then the system H reads as(Jt-J0Δx)z=Hz(t,x,z),(t,x)R×Ω.Assuming H(t,x,z) depends periodically on t with period, say, T>0, we are interested in existence of periodic solutions. There are several papers (cf. [3], [4], [5], [6], [7] and reference therein) devoted to the study on existence of periodic solutions of unbounded Hamiltonian systems similar to (H) under various nonlinear assumptions. The nonlinearity in our situation is more general than that used in [3], [4], [5], [7], and the energy functionals do not satisfy Palais-Smale condition.

We would like to mention that in order to establish a variational setting for the problem (H) we will precisely describe the eigenvalues and eigenfunctions of the operator Jt-J0Δx. Finally, we point out that the result of Theorem 1.1 can be extended to the case where the linear part can be represented as Jt+J0(-Δx+a(x)) with aC(Ω¯).

Let T>0 and set Q[0,T]×Ω, H(t,x,z)12Hz(t,x,z)·z-H(t,x,z). Give the following assumptions:

  • (H1)

    HC1(Q×R2,R) is T-periodic in t;

  • (H2)

    H(t,x,0)0 and |Hz(t,x,z)|=o(|z|) as z0 uniformly in t and x;

  • (H3)

    H(t,x,z)/|z|2 as |z| uniformly in t and x;

  • (H4)

    there is R1>0 such that

    • (1)

      H(t,x,z)a1|z|2 if |z|R1;

    • (2)

      |Hz(t,x,z)||z|σa2H(t,x,z) if |z|R1, where a1,a2>0,σ>1 if N=1 and σ>N2+1 if N2.

Example 1

Hz(t,x,z)=a(t,x)|z|ln(1+|z|) where a(t,x)>0 is T- periodic in t.

Theorem 1.1

Assume (V1)(V2) and (H1)-(H4) hold. Then (H) has at least one nontrivial T-periodic solution.

Section snippets

Preliminaries

In this section we discuss the operator A=(Jt-J0Δx). SettingL2L2(Q,R2),Az(Jt-J0Δx)z,zL2.Since the adjoint of t-x is -t-x, A is selfadjoint operator and σ(A)R.

The nonlinear eigenvalue problem-Δu=λu,xΩ,λR,u=0,xΩ,has eigenvalues 0<μ1<μ2μ3μn,μn+ and eigenfunction φnH01(Ω) corresponding to μn.

Here and in what follows, setting T=2π. Operator Jt has eigenvalues kZ and corresponding eigenfunctions e-kJtc,cR2(cf. [8]) wheree-kJtcL2(S1,R2)=z:S1R2,S1|z|2<,z(t+2π)=z(t),S1=R/(2π

Functional setting

SettingX1X+=L+X=span¯{ψ1,ψ2,ψ3,,ψn,}X2X-=L-X=span¯{ψ-1,ψ-2,ψ-3,,ψ-n,}Xn1span{ψ1,ψ2,ψ3,,ψn},Xn2:=span{ψ-1,ψ-2,ψ-3,,ψ-n}X11X21X1,X12X22X2.For every multi-index α=(α1,α2)N2, we denote by Xα the space Xα11Xα22, we denote by Iα the functional I restricted to Xα.

Lemma 3.1

cf. [1]

Suppose that IC1(X,R) satisfies the following assumptions:

  • (I1)

    I has a local linking at 0,

  • (I2)

    I satisfies (C),

  • (I3)

    I maps bounded sets into bounded sets,

  • (I4)

    for every mN,I(u)-,u,uXm1X2

Then I has at least two critical points.

Lemma 3.2

ΦC1(X

Acknowledgment

The authors thank the reviewer for his (or her) helpful comments.

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