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Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems
Advances in Difference Equations volume 2013, Article number: 242 (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.
Consider the second-order self-adjoint discrete Hamiltonian system
where , , is the forward difference operator, and , is continuously differentiable in x for every . In general, system (1.1) may be regarded as a discrete analogue of the following second-order Hamiltonian system
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:
(AR) there exists such that
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.
In this paper, we are interested in the case when is not global positive definite and satisfies the following assumption.
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 .
(W1) is continuously differentiable in x for every , , and there exist constants and such that
(W2) for all , and
(W3) , ;
(W4) , , where , and there exists such that
(W5) there exist and such that
(W6) there exists a such that
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.
Remark 1.4 In our theorems, is allowed to be sign-changing, for example,
Moreover, is also allowed to be sign-changing. Even if , assumptions (W2), (W4), (W5) and (W6) are weaker than the superquadratic conditions, obtained in the aforementioned references. It is easy to check that the following functions W satisfy (W1), (W2), (W3) and (W4) or (W6):
where , , , , and . One can see that they do not satisfy (W0) or (AR).
Throughout this section, we always assume that is real symmetric positive definite matrix for all . Set
and make the following assumption on instead of (L):
(L′) is an real symmetric matrix for all , , and
and for , let
Then E is a Hilbert space with the inner product above, and the corresponding norm is
As usual, for , set
and their norms are defined by
respectively. Evidently, E is continuously embedded into for , i.e., there exists such that
Lemma 2.1 (Lin and Tang )
For , one has
Lemma 2.2 (Tang and Lin )
Suppose that L satisfies (L′). Then E is compactly embedded in for , and
Now, we define a functional Φ on E by
For any , there exists an such that for . Hence, by (W1), one has
Consequently, under assumptions (L′) and (W1), the functional Φ is of class . Moreover,
Lemma 2.3 Under assumptions (L′), (W1), (W2) and (W4), any sequence satisfying
is bounded in E.
Proof To prove the boundedness of , arguing by contradiction, suppose that . Let . Then and for . Observe that for k large
It follows from (2.7) and (2.9) that
For , let
Passing to a subsequence, we may assume that in E, then by Lemma 2.2, in , , and for all .
If , then in , , for all . Hence, it follows from (W1) that
By virtue of (W4) and (2.10), one can get that
Combining (2.13) with (2.14), we have
which contradicts (2.11).
Set . If , then . For any , we have . Hence for large , and it follows from (2.7), (W1), (W2) and Fatou’s lemma that
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.
Proof Lemma 2.3 implies that is bounded in E. Going if necessary to a subsequence, we can assume that in E. By Lemma 2.2, in for , and for all . By (L′), there exists an integer such that
It is easy to see that
Next, we prove that
If (2.18) is not true, then there exist a constant and a subsequence such that
Since in , passing to a subsequence if necessary, it can be assumed that . Set
Then . From (2.16), (2.20) and (W1), one has
Since for all , then by (2.21), (2.22) and Lebesgue’s dominated convergence theorem, we have
which contradicts (2.19). Hence (2.18) holds. Combining (2.17) with (2.18), one has
It is clear that
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.
Proof First, we prove that is bounded in E. To this end, arguing by contradiction, suppose that . Let . Then and for . By (2.7), (2.8), (2.9) and (W5), one has
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. □
Lemma 2.6 Under assumptions (L′), (W1) and (W2), for any finite-dimensional subspace , there holds
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). □
Corollary 2.7 Under assumptions (L′), (W1) and (W2), for any finite-dimensional subspace , there is such that
Let is an orthonomormal basis of E and define ,
Lemma 2.8 Under assumption (L′), for ,
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]. □
Applying Lemma 2.8, we can choose an integer such that
Lemma 2.9 Under assumptions (L′) and (W1), there exist constants such that .
Proof If , then . Hence, it follows from (W1) that
By (2.7), (2.32) and (2.33), we have
We say that satisfies -condition if any sequence 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
By (L), there exists a constant such that
Let and . Then it is easy to verify the following lemma.
Lemma 3.1 is a solution of system (1.1) if and only if it is a solution of the following system
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. □
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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).
The authors declare that they have no competing interests.
All authors jointly worked on the results, and they read and approved the final manuscript.