In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.
Consider the second-order Hamiltonian systems
where is also T-periodic and satisfies the following assumption (A):
(A) is measurable in t for all , continuously differentiable in u for a.e. and there exist and such that
for all and a.e. .
Here and in the sequel, and always denote the standard inner product and the norm in respectively.
There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [1-7] and the references therein). In , Zhang and Liu studied the asymptotically quadratic case of under the following assumptions:
(AQ1) for all , and there exist constants and such that
(AQ2) uniformly for , and there exist constants such that
(AQ3) uniformly for .
They obtained the existence of infinitely many periodic solutions of (1.1) provided is even inu (see Theorem 1.1 of ).
The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example,  of Jiang and Tang. This paper considers the case of , then . Motivated by  and , we replace (AQ1) with the following condition:
( ) for all , and
The condition ( ) implies that for some constant ,
By the assumption (A) and the condition ( ), for any , there exists a such that
for and a.e. .
Meanwhile, we weaken the condition (AQ3) to ( ) as follows:
( ) There exists a constant such that
Then our main result is the following theorem.
Theorem 1.1Assume that ( ), (AQ2), ( ) hold and is even inu. Then (1.1) possesses infinitely many solutions.
Remark The conditions (AQ1) and (AQ3) are stronger than ( ) and ( ). Then Theorem 1.1 above is different from Theorem 1.1 of .
In this section, we establish the variational setting for our problem and give the variant fountain theorem. Let be the usual Sobolev space with the inner product
We define the functional on E by
where . Then Φ and Ψ are continuously differentiable and
Define a self-adjoint linear operator by
with the domain . Then ℬ has a sequence of eigenvalues ( ). Let be the system of eigenfunctions corresponding to , it forms an orthogonal basis in . Denote by , , it is well known that
and E possesses orthogonal decomposition . For , we have
We can define on E a new inner product and the associated norm by
Therefore, Φ can be written as
Direct computation shows that
for all with and respectively. It is known that is compact.
Denote by the usual norm of , then there exists a such that
We state an abstract critical point theorem founded in . Let E be a Banach space with the norm and with for any . Set and . Consider the following -functional defined by
Theorem 2.1 [, Theorem 2.2]
Assume that the functional defined above satisfies the following:
(T1) maps bounded sets to bounded sets uniformly for , and for all ;
(T2) for all , and as on any finite-dimensional subspace ofE;
(T3) There exist such that
Then there exist , such that
Particularly, if has a convergent subsequence for everyk, then has infinitely many nontrivial critical points satisfying as .
In order to apply this theorem to prove our main result, we define the functionals A, B and on our working space E by
for all and . Then for all . Let , . Note that , where Φ is the functional defined in (2.1).
3 Proof of Theorem 1.1
We firstly establish the following lemmas.
Lemma 3.1Assume that ( ) and ( ) hold. Then for all and as on any finite-dimensional subspace of E.
Proof Since , by (2.4), it is obvious that for all .
By the proof of Lemma 2.6 of , for any finite-dimensional subspace , there exists a constant such that
where is the Lebesgue measure.
For the ϵ given in (3.1), let
Then . By ( ), there exists a constant such that
where is the constant given in (1.2). Note that
for any with . Thus,
for any with . This implies as on Y. □
Lemma 3.2Assume that ( ), (AQ2) and ( ) hold. Then there exist a positive integer and two sequences as such that
where and for all .
Proof Comparing this lemma with Lemma 2.7 of , we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of . We can prove our lemma by using the same method of , so the details are omitted. □
Now it is the time to prove our main result Theorem 1.1.
Proof of Theorem 1.1 By virtue of (1.3), (2.3) and (2.5), maps bounded sets to bounded sets uniformly for . Obviously, for all since is even in u. Consequently, the condition (T1) of Theorem 2.1 holds. Lemma 3.1 shows that the condition (T2) holds, whereas Lemma 3.2 implies that the condition (T3) holds for all , where is given there. Therefore, by Theorem 2.1, for each , there exist and such that
For the sake of notational simplicity, in the following we always set for all .
Step 1. We firstly prove that is bounded in E.
Since satisfies (3.7), one has
Now, we prove that is bounded. Otherwise, without loss of generality, we may assume that
Put , we have . Going to a subsequence if necessary, we may assume that
By (1.3), we have
where . Therefore, one obtains
Passing to the limit in the inequality, by using and as , we obtain
Thus, on a subset Ω of with positive measure.
By (1.2), we have
and by the assumption (A), we obtain
where . So, we get
for all and a.e. . Hence,
An application of Fatou’s lemma yields
which is a contradiction to (3.8).
Step 2. We prove that has a convergent subsequence in E.
Since is bounded in E, E is reflexible and , without loss of generality, we assume
for some .
where is the orthogonal projection for all , that is,
In view of the compactness of and (3.9), the right-hand side of (3.10) converges strongly in E and hence in E. Together with (3.9), we have in E.
Now, from the last assertion of Theorem 2.1, we know that has infinitely many nontrivial critical points. The proof is completed. □
The authors declare that they have no competing interests.
HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.
The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).
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