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.
Keywords:homoclinic solutions; critical point theory; Hamiltonian systems; nontrivial solution
Consider the following second-order Hamiltonian system:
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 thatLandWsatisfy the following conditions:
Then (HS) possesses at least one nontrivial homoclinic solution.
In [15-17], the authors considered the case where is subquadratic as . However, there are many functions with subquadratic growth but they do not satisfy the condition (H2) in  and the corresponding conditions in [16,17]. For example,
Theorem 1.2Let the above condition (H1) hold. Moreover, assume that the following conditions hold:
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 .
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
In order to establish our result via the critical point theory, we firstly describe some properties of the space on which the variational associated with (HS) is defined. Like in , let
Then the space E is a Hilbert space with the inner product
So, the lemma is proved. □
Lemma 2.2 ([, Lemma 1])
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 ofI.
3 Proof of Theorem 1.2
Lemma 3.1Under the assumptions of Theorem 1.2, we have
which yields that
By Lemma 2.3 and the Hölder inequality, we obtain that
Lastly, we check that critical points of I are classical solutions of (HS) satisfying and as . We know that , the space of continuous functions q on ℝ such that as . Moreover, if q is one critical point of I, by (6) we have
Lemma 3.2Under the assumptions of Theorem 1.2, Isatisfies the (PS) condition.
Combining (13) and (14), we obtain that
Now, we can give the proof of Theorem 1.2.
The authors declare that they have no competing interests.
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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