Homoclinic orbits for second-order discrete Hamiltonian systems with subquadratic potential
© Lin; licensee Springer 2013
Received: 3 April 2013
Accepted: 11 July 2013
Published: 31 July 2013
Under the assumptions that is indefinite sign and subquadratic as and satisfies
for some constant , we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system
where and are real symmetric matrices for all , and is always positive definite.
MSC:39A11, 58E05, 70H05.
where , , is the forward difference, and , is continuously differentiable in x for every .
As usual, we say that a solution of system (1.1) is homoclinic (to 0) if as . In addition, if then is called a nontrivial homoclinic solution.
The existence and multiplicity of nontrivial homoclinic solutions for problem (1.1) have been extensively investigated in the literature with the aid of critical point theory and variational methods; see, for example, [1–13]. Most of them treat the case where is superquadratic as .
Compared to the superquadratic case, as far as the author is aware, there are a few papers [10, 12, 13] concerning the case where has subquadratic growth at infinity. Specifically,  and  dealt with the existence and multiplicity of homoclinic solutions for (1.1) under the following assumptions on L:
respectively. In the above two cases, since is positive definite, the variational functional associated with system (1.1) is bounded from below, techniques based on the genus properties have been well applied. In particular, Clark’s theorem is an efficacious tool to prove the existence and multiplicity of homoclinic solutions for system (1.1). However, if is not global positive definite on ℤ, the problem is far more difficult as 0 is a saddle point rather than a local minimum of the variational functional, which is strongly indefinite and it is not easy to obtain the boundedness of the Palais-Smale sequence. In a recent paper , based on a new direct sum decomposition of the ‘work space’, Tang and Lin proved the following theorem by using a linking theorem which was developed in .
Theorem 1.1 
Assume that is an real symmetric positive definite matrix for all , L and W satisfy the following assumptions:
where as , ;
(W5) , .
Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.
We remark that the condition ‘positive definite’ is removed in (), i.e., is not required to be global positive definite on ℤ. The main goal of this paper is to weaken conditions (W1), (W2), (W3) and (W4) of Theorem 1.1 under assumption ().
To state our result, we first introduce the following assumptions:
where as , ;
We are now in a position to state the main result of this paper.
Theorem 1.2 Assume that is an real symmetric positive definite matrix for all , L and W satisfy (), (W1′), (W2′), (W3′), (W4′) and (W5). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.
Lemma 2.1 [, Lemma 2.2]
Lemma 2.2 [, Lemma 2.3]
- (i)is a bilinear function on E, and there exists a constant such that(2.8)
This shows that (2.12) holds.
This shows that possesses a convergent subsequence in . Therefore, E is compactly embedded in for . □
This shows that (2.19) holds. □
Furthermore, the critical points of f in E are the solutions of system (1.1) with .
This shows that (2.24) holds. In view of the proof of [, Lemma 2.6], the critical points of f in E are the solutions of system (1.1) with . □
which implies the continuity of . The proof is complete. □
Lemma 2.6 
f is bounded from below on ;
for each finite dimensional subspace , there are positive constants and such that and , where .
Then f possesses infinitely many nontrivial critical points.
3 Proof of the theorem
Since , it follows from (3.8), (3.13) and (3.14) that in E. Hence, f satisfies the (PS)-condition.
as and , since .
This shows that assumption (ii) in Lemma 2.6 holds. By Lemma 2.6, f has infinitely many critical points which are homoclinic solutions for system (1.1). □
In this section, we give an example to illustrate our result.
Hence, by Theorem 1.2, system (1.1) has infinitely many nontrivial homoclinic solutions. However, one can see that defined by (4.1) does not satisfy (W1) and (W2).
The author would like to express their thanks to the referees for their helpful suggestions. This work is partially supported by the NNSF (No. 11171351) of China and supported by the Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by the Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).
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