Periodic solutions of an infinite-dimensional Hamiltonian system☆
Introduction
Based on a developed local linking theorem (cf. [1], [2]) we look for periodic solutions of the following supquadratic Hamiltonian system:where is a bounded domain with smooth boundary and is a function. Settingthen the system reads asAssuming depends periodically on t with period, say, , 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 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 we will precisely describe the eigenvalues and eigenfunctions of the operator . 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 with .
Let and set , . Give the following assumptions:
is T-periodic in t;
and as uniformly in t and x;
as uniformly in t and x;
there is such that
- (1)
if ;
- (2)
if , where if and if .
- (1)
Example 1
where is T- periodic in t.
Theorem 1.1
Assume and hold. Then has at least one nontrivial T-periodic solution.
Section snippets
Preliminaries
In this section we discuss the operator . SettingSince the adjoint of is , A is selfadjoint operator and .
The nonlinear eigenvalue problemhas eigenvalues and eigenfunction corresponding to .
Here and in what follows, setting . Operator has eigenvalues and corresponding eigenfunctions (cf. [8]) where
Functional setting
SettingFor every multi-index , we denote by the space , we denote by the functional I restricted to . Lemma 3.1 Suppose that satisfies the following assumptions: I has a local linking at 0, I satisfies , I maps bounded sets into bounded sets, for every cf. [1]
Then I has at least two critical points.
Lemma 3.2
Acknowledgment
The authors thank the reviewer for his (or her) helpful comments.
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Supported by NSFC 10471075,10671195,10771117,10726004 and Q2007A02.