Abstract
By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic secondorder 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 solution1 Introduction
Consider the following secondorder Hamiltonian system:
where , is a symmetric matrixvalued 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 secondorder Hamiltonian systems have been extensively investigated in many papers (see, e.g., [117] and the references therein) via variational methods. More precisely, many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [517]. Some of them treated the case where and are either independent of t or periodic in t (see, for instance, [57]), and a more general case is considered in the recent paper [7]. 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 [8], many results [917] were obtained for the case where is neither constant nor periodic in t.
Recently, Zhang and Yuan [15] 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 [[15], Theorem 1.1])
Assume thatLandWsatisfy the following conditions:
(H1) is a symmetric matrix for all, and there is a continuous functionsuch thatfor allandandas.
(H2) whereis a positive continuous function such thatandis a constant.
Then (HS) possesses at least one nontrivial homoclinic solution.
In [1517], 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 [15] and the corresponding conditions in [16,17]. For example,
where , are positive continuous functions such that .
In this paper, our aim is to revisit (HS) and study the subquadratic case which is not included in [1517]. Now, we state our main result.
Theorem 1.2Let the above condition (H1) hold. Moreover, assume that the following conditions hold:
(H3) , , whereis a positive continuous function such thatandis a constant.
(H4) , whereare 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 [15].
Remark 1.2 We still consider the function defined in (1),
Due to , there are no constants such that
so does not satisfy the conditions (W2) and (W3) in [16]. Moreover, for any given , does not satisfy the condition (W2) in [17]. Therefore, we also extend Theorem 1.2 in [16] and Theorem 1.1 in [17].
Example 1.1 Consider the following secondorder Hamiltonian system with :
where
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 [17], then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in [17].
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 [15], let
Then the space E is a Hilbert space with the inner product
and the corresponding norm . Note that
for all with the embedding being continuous. Here () and denote the Banach spaces of functions on ℝ with values in under the norms
and
respectively. In particular, for , there exists a constant such that
Lemma 2.1There exists a constantsuch that if, then
Proof From (H1), we can imply that there exists a constant such that
for all and . By the above inequality, one has
So, the lemma is proved. □
Lemma 2.2 ([[9], Lemma 1])
Suppose thatLsatisfies (H1). Then the embedding ofEinis compact.
Lemma 2.3Suppose that (H1) and (H4) are satisfied. If (weakly) inE, thenin.
Proof Assume that in E. Then there exists a constant such that, by the BanachSteinhaus theorem and (3),
Since , by (H4) there exists a constant such that
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 Fréchetdifferentiable 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 ([[18], Theorem 2.7])
LetEbe a real Banach space, and let us havesatisfying the (PS) condition. IfIis bounded from below, then
is a critical value ofI.
3 Proof of Theorem 1.2
Now, we are going to establish the corresponding variational framework to obtain homoclinic solutions of (HS). Define the functional
Lemma 3.1Under the assumptions of Theorem 1.2, we have
which yields that
Moreover, Iis a continuously Fréchetdifferentiable functional defined onE, i.e., and any critical point ofIonEis a classical solution of (HS) with.
Proof We firstly show that . Let , by (3), (H4), and the Hölder inequality, we have
Combining (5) and (8), we show that . Next, we prove that . Rewrite I as follows:
where
Thus, it is sufficient to show that this is the case for . In the process we will see that
which is defined for all . For any given , let us define as follows:
It is obvious that is linear. Now, we show that is bounded. Indeed, for any given , by (3) and (H4), there exists a constant such that
for all , which yields that by (4) and the Hölder inequality,
Moreover, for any , by the mean value theorem, we have
where . Therefore, by Lemma 2.3 and the Hölder inequality, one has
as in E. Combining (11) and (12), we see that (10) holds. It remains to prove that is continuous. Suppose that in E and note that
By Lemma 2.3 and the Hölder inequality, we obtain that
as , which implies the continuity of and .
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
which yields that , i.e., q is a classical solution of (HS). Since q is one critical point of I, we have
It follows from as and the above equality that
Hence, q satisfies as . This proof is complete. □
Lemma 3.2Under the assumptions of Theorem 1.2, Isatisfies the (PS) condition.
Proof In fact, assume that is a sequence such that is bounded and as . Then there exists a constant such that
We firstly prove that is bounded in E. By (5) and (8), we have
Combining (13) and (14), we obtain that
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
Hence,
as . Moreover, an easy computation shows that
So, as , i.e., I satisfies the PalaisSmale condition. □
Now, we can give the proof of Theorem 1.2.
Proof of Theorem 1.2 By (5) and (8), for every and , we have
Since , (16) implies that as . Consequently, I is a functional bounded from below. By Lemmas 3.2 and 2.4, I possesses a critical value , i.e., there is a such that
On the other hand, take with , and let be given by
which yields that as small enough since , i.e., the critical point obtained above is nontrivial. □
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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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