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Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential
- Xiaoyan Lin1Email author and
- Xianhua Tang2
https://doi.org/10.1186/1687-1847-2013-154
© Lin and Tang; licensee Springer 2013
- Received: 3 October 2012
- Accepted: 10 May 2013
- Published: 30 May 2013
Abstract
In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system
Under the assumption that is of indefinite sign and subquadratic as and and are real symmetric positive definite matrices for all , and that
for some constant , we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark’s theorem in critical point theory.
MSC:39A11, 58E05, 70H05.
Keywords
- homoclinic solution
- discrete Hamiltonian system
- critical point
- Clark’s theorem
1 Introduction
where , , is the forward difference operator, and . 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.
Moreover, system (1.1) does have its applicable setting as evidenced by monographs [1, 2]. System (1.2) can also be regarded as a special form of the Emden-Fowler equation appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting system, and many well-known results concerning properties of solutions of (1.2) are collected in [3].
or other superquadratic growth conditions, where and in the sequel, denotes the standard inner product in and is the induced norm.
When is of subquadratic growth at infinity, Tang and Lin [9] recently established the following results on the existence of homoclinic solutions of system (1.1).
Theorem A [9]
- (L)is an real symmetric positive definite matrix for all , and there exists a constant such that
where as ;
Then system (1.1) possesses at least one nontrivial homoclinic solution.
Theorem B [9]
Assume that is an real symmetric positive definite matrix for all , and that L and W satisfy (L), (W1), (W2) and the following assumptions:
(W5) , .
Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.
When satisfies (L) in Theorems A and B, assumption (W1) is optimal in some sense, essentially, the summable functions are necessary; see Lemma 2.2 in Section 2.
Now a natural question is whether the conditions on the potential can be further relaxed when one imposes stronger conditions on ?
In the present paper, we give a positive answer to the above question. In fact, we employ Clark’s theorem in critical point theory to establish new existence criteria to guarantee that system (1.1) has at least one or multiple homoclinic solutions under the following assumption instead of (L):
Our main results are the following two theorems.
Theorem 1.1 Assume that is an real symmetric positive definite matrix for all , that L satisfies (L ν ) and W satisfies the following assumptions:
where as , ;
Then system (1.1) possesses at least one nontrivial homoclinic solution.
Theorem 1.2 Assume that is an real symmetric positive definite matrix for all , and that L and W satisfy (L ν ), (W1′), (W2′), (W5) and the following assumption:
Then system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions.
Remark 1.3 Obviously, assumptions (W1′), (W2′), (W3′) and (W4′) are weaker than (W1), (W2), (W3) and (W4), respectively.
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our theorems. In Section 4, we give some examples to illustrate our results.
2 Preliminaries
respectively.
Lemma 2.1 [9]
where and .
This shows that (2.4) holds.
This shows that possesses a convergent subsequence in . Therefore, E is compactly embedded in for . □
This shows (2.12) holds. □
Furthermore, the critical points of f in E are solutions of (1.1) with .
So, 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 2.5 [10]
Let E be a real Banach space and let satisfy the (PS)-condition. If f is bounded from below, then is a critical value of f.
Lemma 2.6 [11]
Let E be a real Banach space, let with f even, bounded from below, and satisfying the (PS)-condition. Suppose that , there is a set such that K is homeomorphic to by an odd map, and . Then f possesses at least k distinct pairs of critical points.
3 Proofs of theorems
Since , (3.2) implies that as . Consequently, f is bounded from below.
Since , it follows from (3.10) and (3.11) that in E. Hence, f satisfies the (PS)-condition.
By Lemma 2.5, is a critical value of f, that is, there exists a critical point such that .
Since , it follows from (3.12) that for small enough. Hence , therefore is a nontrivial critical point of f, and so is a nontrivial homoclinic solution of system (1.1). The proof is complete. □
and so . By Lemma 2.4, f has at least m distinct pairs of critical points, and so system (1.1) possesses at least m distinct pairs of nontrivial homoclinic solutions. The proof is complete. □
4 Examples
In this section, we give three examples to illustrate our results.
By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.
By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.
By Theorem 1.2, system (1.1) has at least m distinct pairs of nontrivial homoclinic solutions. Since m is arbitrary, it follows that system (1.1) has infinitely many distinct pairs of nontrivial homoclinic solutions.
Declarations
Acknowledgements
The authors 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 Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).
Authors’ Affiliations
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