Homoclinic orbits for second order Hamiltonian systems with asymptotically linear terms at infinity
© Chen; licensee Springer 2014
Received: 16 March 2014
Accepted: 14 April 2014
Published: 2 May 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.
Keywordshomoclinic orbits second order Hamiltonian systems asymptotically linear variational method
1 Introduction and main result
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 .
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.
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 ()
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.
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 .
which implies the conclusion. □
Lemma 2.2 There is a function with satisfying .
Therefore, we can choose with s big enough such that with satisfying . □
An application of Theorem A now completes the proof. □
Therefore, the proof follows from the definition of G. □
then up to a subsequence, with and .
It implies that .
Therefore, the proof is finished. □
Lemma 2.6 The sequence obtained in (2.8) is bounded.
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.
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. □
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).
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