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Homoclinic orbits for second order Hamiltonian systems with asymptotically linear terms at infinity
Advances in Difference Equations volume 2014, Article number: 114 (2014)
In this paper, by using some different asymptotically linear conditions from those previously used in Hamiltonian systems, we obtain the existence of nontrivial homoclinic orbits for a class of second order Hamiltonian systems by the variational method.
MSC:37J45, 37K05, 58E05.
1 Introduction and main result
We consider the following second order Hamiltonian system:
where is T-periodic in t, denotes its gradient with respect to the x variable, and is the T-periodic matrix that satisfies
and it is symmetric and positive definite uniformly for . We say that a solution of (1.1) is homoclinic (with 0) if such that and as . If , then is called a nontrivial homoclinic solution.
Let . We assume:
(H1) is T-periodic in t, and , .
(H2) There are some constants and such that if .
(H3) There is a constant such that
(H4) , , and there exist , , and such that
Now, our main result reads as follows.
Theorem 1.1 If (1.2) and (H1)-(H4) with hold, then (1.1) has a nontrivial homoclinic orbit.
Example 1.1 Let
where is T-periodic in t, and . It is not hard to check that the above function satisfies (H1)-(H4).
We will use the following theorem to prove our main result.
Theorem A ()
Let E be a Banach space equipped with the norm and let be an interval. We consider a family of -functionals on E of the form
where , and such that either or as . We assume there are two points in E such that setting
we have, ,
Then, for almost every , there is a sequence such that
In recent decades, many authors are devoted to the existence and multiplicity of homoclinic orbits for second order Hamiltonian systems with super or asymptotically linear terms by critical point theory, see [2–22] and the references therein. Many authors [2–7, 9, 11–13, 15, 17–19] have studied the existence of homoclinic orbits of (1.1) by considering the following so-called global Ambrosetti-Rabinowitz condition on H due to Ambrosetti and Rabinowitz (e.g., ): there exists a constant such that
where denotes the inner product in , and the corresponding norm is denoted by . Roughly speaking the role of (1.3) is to insure that all Palais-Smale sequences for the corresponding function of (1.1) at the mountain-pass level are bounded. By removing or weakening the condition (1.3), some authors studied the homoclinic orbits of (1.1). For example, Zou and Li  proved that the system (1.1) has infinitely many homoclinic orbits by using the variant fountain theorem; Chen  obtained the existence of a ground state homoclinic orbit for (1.1) by a variant generalized weak linking theorem due to Schechter and Zou. Ou and Tang  obtained the existence of a homoclinic solution of (1.1) by the minimax methods in the critical point theory. For second order Hamiltonian systems without periodicity, we refer the readers to [20–22] and so on.
The rest of our paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proof of our result. In Section 3, we give the detailed proof of our result.
2 Preliminary lemmas
Throughout this paper we denote by the usual norm and C for generic constants.
In what follows, we always assume (1.2) and (H1)-(H4) with hold. Let under the usual norm
Thus E is a Hilbert space and it is not difficult to show that , the space of continuous functions u on ℝ such that as (see, e.g., ). We will seek solutions of (1.1) as critical points of the functional I associated with (1.1) and given by
We define a new norm
and its corresponding inner product is denoted by . By (1.2), can and will be taken as an equivalent norm on E. Hence I can be written as
The assumptions on H imply that . Moreover, critical points of I are classical solutions of (1.1) satisfying as . Thus u is a homoclinic solution of (1.1). Let us show that I has a mountain-pass geometry. Since this is a consequence of the two following results.
Lemma 2.1 as .
Proof By (H2) and (H3), we know for any there exists a such that
where . It follows from (see (H4)) that
By (2.3) and the Sobolev embedding theorem, we deduce that
which implies the conclusion. □
Lemma 2.2 There is a function with satisfying .
Obviously, , . Straightforward calculations show that
For every , as . It follows from (H3) that
which together with (2.4), the definition of (above Theorem 1.1) and the Fatou lemma implies
Therefore, we can choose with s big enough such that with satisfying . □
We define on E the family of functionals
Lemma 2.3 The family with satisfies the hypotheses of Theorem A. In particular for almost every there is a bounded sequence satisfying
Proof For the obtained in Lemma 2.2, we have . It follows from (H1) that , . By the proof in Lemma 2.1, we have
then it follows from (2.5) and (2.6) that
An application of Theorem A now completes the proof. □
Lemma 2.4 If vanishes and is bounded, then
Proof It is known that if vanishes, then in for all , which together with (2.2), (2.3), and the Hölder inequality implies
Therefore, the proof follows from the definition of G. □
Lemma 2.5 If is bounded in E and satisfies
then up to a subsequence, with and .
Proof Note that is bounded and
it follows from Lemma 2.4 that does not vanish, i.e., there are and a sequence such that
It implies that .
Observe that (H4) implies for all , which together with the Fatou lemma and implies
Therefore, the proof is finished. □
By Lemmas 2.3 and 2.5, we deduce the existence of a sequence such that:
Clearly is increasing and bounded by , and it follows that
Lemma 2.6 The sequence obtained in (2.8) is bounded.
Proof Since is T-periodic in t, by (H4) and (H3), respectively,
for some positive constants and . Note that (2.8) implies
thus it follows from (H4) and (2.10) that
Take , then by (2.12), the Hölder inequality, and the Sobolev imbedding theorem,
for some positive constant , where . Note that , it follows from (H2), (2.11)-(2.13), the Hölder inequality and the Sobolev imbedding theorem that
for some positive constants , , and , where . Therefore, (2.14) implies that is bounded and the proof is finished. □
3 Proof of main result
We are now in a position to prove our main result.
Proof of Theorem 1.1 is bounded by Lemma 2.6, so we can assume and a.e. , up to a subsequence. By (2.8), we have
We distinguish two cases: either or . If the first case holds, we get and the result follows from Lemma 2.5. If , we define the sequence by with satisfying
(If for a , defined by (3.1) is not unique we choose the smaller possible value.) Since is bounded, is bounded. Note that , , thus
On the other hand it is easily seen, following the proof of Lemma 2.1, that as , uniformly in . Therefore, since , there is such that , . Recording that , then we obtain from Lemma 2.1 and (3.1) , and from (3.2) it follows that
It follows from the fact is bounded and Lemma 2.4 that does not vanish, so does not vanish. The proof of and is similar to the proof of Lemma 2.5. □
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Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).
The author declares that they have no competing interests.
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Cite this article
Chen, G. Homoclinic orbits for second order Hamiltonian systems with asymptotically linear terms at infinity. Adv Differ Equ 2014, 114 (2014). https://doi.org/10.1186/1687-1847-2014-114
- homoclinic orbits
- second order Hamiltonian systems
- asymptotically linear
- variational method