Existence of homoclinic solutions for a class of difference systems involving p-Laplacian
© Zhang; licensee Springer. 2014
Received: 20 August 2014
Accepted: 10 November 2014
Published: 25 November 2014
By using the critical point theory, some existence criteria are established which guarantee that the difference p-Laplacian systems of the form have at least one or infinitely many homoclinic solutions, where , , , , , and are not periodic in n.
MSC:34C37, 35A15, 37J45, 47J30.
Keywordshomoclinic solutions variational methods difference p-Laplacian systems
Problem (1.2) has been studied by Shi et al. in  and problem (1.3) has been studied in [2–4]. It is well known that the existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been firstly recognized by Poincaré . If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. Therefore, it is of practical importance to investigate the existence of homoclinic orbits of (1.1) emanating from 0.
By applying critical point theory, the authors [6–22] studied the existence of periodic solutions and subharmonic solutions for difference equations or differential equations, which show that the critical point theory is an effective method to study periodic solutions of difference equations or differential equations. In this direction, several authors [23–34] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated mainly by the ideas of [1–4, 35], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and the symmetric mountain pass theorem. More precisely, we obtain the following main results, which seem not to have been considered in the literature.
- (A)Let and , is a positive function on ℤ such that for all
uniformly in .
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.2 Suppose that a and W satisfy (A), (W2) and the following conditions:
uniformly in .
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.3 Suppose that a and W satisfy (A), (W1)-(W3) and
(W4) , .
Then problem (1.1) has an unbounded sequence of homoclinic solutions.
Theorem 1.4 Suppose that a and W satisfy (A), (W1)′, (W2), (W3)′ and (W4). Then problem (1.1) has an unbounded sequence of homoclinic solutions.
The rest of this paper is organized as follows: in Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.
is a reflexive Banach space.
Furthermore, the critical points of φ in E are classical solutions of (1.1) with .
Lemma 2.1 
where from (A). Then (2.4) holds.
(2.5) holds by taking .
Finally, as is the weighted Sobolev space , it follows from  that (2.6) holds. □
The following two lemmas are the mountain pass theorem and the symmetric mountain pass theorem, which are useful in the proofs of our theorems.
Lemma 2.3 
There exist constants such that .
There exists an such that .
where , and is an open ball in E of radius ρ centered at 0.
Lemma 2.4 
For each finite dimensional subspace , there is such that for , is an open ball in E of radius r centered at 0.
Then I possesses an unbounded sequence of critical values.
is nondecreasing on ;
is nonincreasing on .
The proof of Lemma 2.5 is routine and we omit it. In the following, () denote different positive constants.
3 Proofs of theorems
Hence, we have in E by (3.15) and (3.16). This shows that φ satisfies the (PS)-condition.
Therefore, we can choose a constant depending on ρ such that for any with .
Since and , it follows from (3.24) that there exists such that and . Let , then , , and .
The function is a desired solution of problem (1.1). Since , is a nontrivial homoclinic solution. The proof is complete. □
Therefore, we can choose a constant depending on ρ such that for any with . The proof of Theorem 1.2 is complete. □
This contradicts the fact that is unbounded, and so is unbounded. The proof is complete. □
Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □
Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with and . Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.
Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with and . Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.
This work was supported by the NNSF of China (No. 11301108), Guangxi Natural Science Foundation (No. 2013GXNSFBA019004) and the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093).
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