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Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

Hua Gu* and Tianqing An

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College of Science, Hohai University, Nanjing, 210098, China

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Boundary Value Problems 2013, 2013:16  doi:10.1186/1687-2770-2013-16

 Received: 8 November 2012 Accepted: 10 January 2013 Published: 6 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1 Introduction

Consider the second-order Hamiltonian systems

(1.1)

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 [6], 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 [6]).

The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example, [4] of Jiang and Tang. This paper considers the case of , then . Motivated by [4] and [6], we replace (AQ1) with the following condition:

() for all , and

The condition () implies that for some constant ,

(1.2)

By the assumption (A) and the condition (), for any , there exists a such that

(1.3)

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 andis 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 [6].

2 Preliminaries

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

and

Therefore, Φ can be written as

(2.1)

Direct computation shows that

(2.2)

for all with and respectively. It is known that is compact.

Denote by the usual norm of , then there exists a such that

(2.3)

We state an abstract critical point theorem founded in [8]. Let E be a Banach space with the norm and with for any . Set and . Consider the following -functional defined by

Theorem 2.1 [[8], Theorem 2.2]

Assume that the functionaldefined above satisfies the following:

(T1) maps bounded sets to bounded sets uniformly for, andfor all;

(T2) for all, andason any finite-dimensional subspace ofE;

(T3) There existsuch that

and

Then there exist, such that

Particularly, ifhas a convergent subsequence for everyk, thenhas infinitely many nontrivial critical pointssatisfyingas.

In order to apply this theorem to prove our main result, we define the functionals A, B and on our working space E by

(2.4)

and

(2.5)

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. Thenfor allandason 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 [6], for any finite-dimensional subspace , there exists a constant such that

(3.1)

where is the Lebesgue measure.

For the ϵ given in (3.1), let

Then . By (), there exists a constant such that

(3.2)

where is the constant given in (1.2). Note that

(3.3)

for any with . Thus,

for any with . This implies as on Y. □

Lemma 3.2Assume that (), (AQ2) and () hold. Then there exist a positive integerand two sequencesassuch that

(3.4)

(3.5)

and

(3.6)

whereandfor all.

Proof Comparing this lemma with Lemma 2.7 of [6], we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of [6]. We can prove our lemma by using the same method of [6], 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

(3.7)

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

More precisely,

(3.8)

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

(3.9)

for some .

Note that

where is the orthogonal projection for all , that is,

(3.10)

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. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.

Acknowledgements

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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