Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth
© Zhang; licensee Springer 2012
Received: 6 July 2012
Accepted: 25 October 2012
Published: 13 November 2012
By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.
where , is a symmetric matrix-valued function, and , is the gradient of W about q. As usual we say that a solution of (HS) is homoclinic (to 0) if such that and as . If , is called a nontrivial homoclinic solution.
By now, the existence and multiplicity of homoclinic solutions for second-order Hamiltonian systems have been extensively investigated in many papers (see, e.g., [1–17] and the references therein) via variational methods. More precisely, many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [5–17]. Some of them treated the case where and are either independent of t or periodic in t (see, for instance, [5–7]), and a more general case is considered in the recent paper . If is neither constant nor periodic in t, the problem of the existence of homoclinic solutions for (HS) is quite different from the one just described due to the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka , many results [9–17] were obtained for the case where is neither constant nor periodic in t.
Recently, Zhang and Yuan  obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument. In this paper, denotes the standard inner product in , and subsequently, is the induced norm. If , then .
Theorem 1.1 (See [, Theorem 1.1])
Assume that L and W satisfy the following conditions:
(H1) is a symmetric matrix for all , and there is a continuous function such that for all and and as .
(H2) where is a positive continuous function such that and is a constant.
Then (HS) possesses at least one nontrivial homoclinic solution.
where , are positive continuous functions such that .
Theorem 1.2 Let the above condition (H1) hold. Moreover, assume that the following conditions hold:
(H3) , , where is a positive continuous function such that and is a constant.
(H4) , where are positive continuous functions such that .
Then (HS) possesses at least one nontrivial homoclinic solution.
Remark 1.1 Obviously, the condition (H2) is a special case of (H3)-(H4). If (H2) holds, so do (H3)-(H4); however, the reverse is not true. defined in (1) can satisfy the conditions (H3) and (H4), but cannot satisfy the condition (H2). So, we generalize and significantly improve Theorem 1.1 in .
Let , and , , , . Clearly, (H1), (H3), and (H4) hold. Therefore, by applying Theorem 1.2, the Hamiltonian system (2) possesses at least one nontrivial homoclinic solution.
Remark 1.3 It is easy to see that (H2) in Theorem 1.1 is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1. On the other hand, W does not satisfy the conditions (W2) and (W5) of , then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in .
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.
2 Preliminary results
So, the lemma is proved. □
Lemma 2.2 ([, Lemma 1])
Suppose that L satisfies (H1). Then the embedding of E in is compact.
Lemma 2.3 Suppose that (H1) and (H4) are satisfied. If (weakly) in E, then in .
On the other hand, by Lemma 2.2, in , passing to a subsequence if necessary, which implies for almost every . Then using Lebesgue’s convergence theorem, the lemma is proved. □
Now, we introduce more notation and some necessary definitions. Let E be a real Banach space, , which means that I is a continuously Fréchet-differentiable functional defined on E. Recall that is said to satisfy the (PS) condition if any sequence , for which is bounded and as , possesses a convergent subsequence in E.
Lemma 2.4 ([, Theorem 2.7])
is a critical value of I.
3 Proof of Theorem 1.2
Moreover, I is a continuously Fréchet-differentiable functional defined on E, i.e., and any critical point of I on E is a classical solution of (HS) with .
as , which implies the continuity of and .
Hence, q satisfies as . This proof is complete. □
Lemma 3.2 Under the assumptions of Theorem 1.2, I satisfies the (PS) condition.
for every .
Since , the above inequality shows that is bounded in E. By Lemma 2.2, the sequence has a subsequence, again denoted by , and there exists such that
, weakly in E,
, strongly in .
So, as , i.e., I satisfies the Palais-Smale condition. □
Now, we can give the proof of Theorem 1.2.
which yields that as small enough since , i.e., the critical point obtained above is nontrivial. □
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
This work is supported by the Research Foundation of Education Bureau of Hunan Province, China (No.11C0594). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
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