Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems
© Tang and Chen; licensee Springer. 2013
Received: 19 April 2013
Accepted: 24 July 2013
Published: 8 August 2013
In the present paper, we deal with 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. Under the assumptions that is allowed to be sign-changing and satisfies
is of indefinite sign and superquadratic as , we establish several existence criteria to guarantee that the above system has infinitely many homoclinic solutions.
MSC:39A11, 58E05, 70H05.
Moreover, system (1.1) has applications as is shown in the monographs [1, 2]. In the past 40 years, system (1.2) has been widely investigated, see [3–9] and references therein. System (1.2) is the special form of the Emden-Fowler equation, appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, and many well-known results concerning properties of solutions of (1.2) are collected in .
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 the multiplicity of homoclinic solutions of system (1.1) or its special forms have been investigated by many authors. Papers [11–13] deal with the periodic case, where p, L and W are N-periodic in n. If the periodicity is lost, the case is quite different from the ones just described, because of lack of compactness of the Sobolev embedding. In this case, either a coercivity condition on L are required to be satisfied, see [14–19], or can be dominated by a summable function, see [5, 13]. In the above-mentioned papers, except , L was always required to be positive definite. Meanwhile, W was always assumed to be superquadratic as uniformly for , i.e.,
(W0) uniformly for .
In addition, is subquadratic as in [17, 20], while is superquadratic in [11–16, 18, 19, 21]. Moreover, in the superquadratic case, except , the well-known global Ambrosetti-Rabinowitz superquadratic condition was always assumed:
where and in the sequel, denotes the standard inner product in , and is the induced norm.
However, in mathematical physics, it is of frequent occurrence in a system like (1.1) that the global positive definiteness of is not satisfied. This is seen, for example, , where , as , and is bounded, or , is a polynomial of degree 2m with the property that the coefficient of the leading term is positive.
- (L)is an real symmetric matrix for all and the smallest eigenvalue of as , i.e.,
Under assumption (L) above, we will use the symmetric mountain pass theorem to study the existence of infinitely many homoclinic solutions for (1.1) in the case, where W satisfies the following weaker assumptions than (W0) as and (AR) as .
(W3) , ;
Now, we are ready to state the main results of this paper.
Theorem 1.1 Assume that is an real symmetric positive definite matrix for all , L and W satisfy (L), (W1), (W2), (W3) and (W4). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.
Theorem 1.2 Assume that is an real symmetric positive definite matrix for all , L and W satisfy (L), (W1), (W2), (W3) and (W5). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.
It is easy to check that (W1) and (W6) imply (W5). Thus, we have the following corollary.
Corollary 1.3 Assume that is an real symmetric positive definite matrix for all , L and W satisfy (L), (W1), (W2), (W3) and (W6). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.
where , , , , and . One can see that they do not satisfy (W0) or (AR).
and make the following assumption on instead of (L):
Lemma 2.1 (Lin and Tang )
Lemma 2.2 (Tang and Lin )
is bounded in E.
Passing to a subsequence, we may assume that in E, then by Lemma 2.2, in , , and for all .
which contradicts (2.11).
which is a contradiction. Thus is bounded in E. □
Lemma 2.4 Under assumptions (L′), (W1), (W2) and (W4), any sequence satisfying (2.9) has a convergent subsequence in E.
From (2.23), (2.24) and (2.25), we have , . □
Lemma 2.5 Under assumptions (L′), (W1) and (W5), any sequence satisfying (2.9) has a convergent subsequence in E.
Passing to a subsequence, we may assume that in E, then by Lemma 2.1, in , , and for all . Hence, it follows from (2.27) that . Analogous to the proof of (2.15), we can deduce a contradiction. Thus, is bounded in E. The rest of the proof is the same as the one in Lemma 2.4. □
Proof Arguing indirectly, assume that for some sequence with , and there is such that for all . Set , then . Passing to a subsequence, we may assume that in E. Since is finite-dimensional, then in E, for all , and so . Hence, we can deduce a contradiction in the same way as (2.15). □
Proof Since the embedding from E into is compact for , then Lemma 2.8 can be proved in a similar way as [, Lemma 3.8]. □
Lemma 2.9 Under assumptions (L′) and (W1), there exist constants such that .
has a convergent subsequence.
Lemma 2.10 (Bartolo, Benci and Fortunato )
Let X be an infinite-dimensional Banach space, , where Y is finite-dimensional. If satisfies -condition for all , and
(I1) , for all ;
(I2) there exist constants such that ;
(I3) for any finite-dimensional subspace , there is such that on ;
then I possesses an unbounded sequence of critical values.
3 Proofs of the main results
Let and . Then it is easy to verify the following lemma.
Proof of Theorem 1.1 Let , , and let . Obviously, satisfies (W1), (W2), (W3) and (W4). By Lemmas 2.3, 2.4, 2.9 and Corollary 2.7, all conditions of Lemma 2.10 are satisfied. Thus, system (3.2) possesses infinitely many nontrivial solutions. By Lemma 3.1, system (1.1) also possesses infinitely many nontrivial solutions. □
Proof of Theorem 1.2 Let , , and let . Obviously, satisfies (W1), (W2), (W3) and (W5). The rest of the proof is the same as that of Theorem 1.1 by using Lemma 2.5 instead of Lemmas 2.3 and 2.4. □
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 the Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).
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