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Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential
Advances in Difference Equations volume 2013, Article number: 154 (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.
1 Introduction
Consider the second-order self-adjoint discrete Hamiltonian system
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.
In general, system (1.1) may be regarded as a discrete analogue of the following second order Hamiltonian system:
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].
In papers [4–8], the authors studied the existence of homoclinic solutions of system (1.1) or some of its special forms under the following superquadratic growth (AR)-condition on W: there is a constant such that
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]
Assume that is an real symmetric positive definite matrix for all , and that L and W satisfy the following assumptions:
-
(L)
is an real symmetric positive definite matrix for all , and there exists a constant such that
(W1) For every , W is continuously differentiable in x, and there exist two constants and two functions such that
and
(W2) There exist two functions and such that
where as ;
(W3) There exist an and two constants and such that
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:
(W4) There exist two constants and and a set with elements such that
(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):
(L ν ) is an real symmetric positive definite matrix for all , and there exists a constant such that
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:
(W1′) There exist constants and such that
(W2′) There exists a function such that
where as , ;
(W3′) There exist and constants and such that
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:
(W4′) There exist constants and and a set with elements such that
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
In this section, we always assume that and are real symmetric positive definite matrices for all . Let
and for , let
Then E is a Hilbert space with the above inner product, and the corresponding norm is
As usual, for , set
and
and their norms are defined by
respectively.
Lemma 2.1 [9]
For ,
where and .
By (L ν ), there exist two constants and such that
which implies
Lemma 2.2 Suppose that L satisfies (L ν ). Then, for , E is compactly embedded in ; moreover,
and
where
and
Proof Let . Then . For and , it follows from (2.2), (2.5) and the Hölder inequality that
This shows that (2.3) holds. Hence, from (2.2), (2.6) and the Hölder inequality, one has
This shows that (2.4) holds.
Finally, we prove that E is compactly embedded in . Let be a bounded sequence. Then by (2.1), there exists a constant such that
Since E is reflexive, possesses a weakly convergent subsequence in E. Passing to a subsequence if necessary, it can be assumed that in E. It is easy to verify that
For any given number , we can choose such that
It follows from (2.8) that there exists such that
On the other hand, it follows from (2.3), (2.7) and (2.9) that
Since ε is arbitrary, combining (2.10) with (2.11), we get
This shows that possesses a convergent subsequence in . Therefore, E is compactly embedded in for . □
Lemma 2.3 Suppose that L and W satisfy (L ν ) and (W1′). Then, for ,
where
Proof For , it follows from (2.4) and (W1′) that
This shows (2.12) holds. □
Lemma 2.4 Assume that L and W satisfy (L ν ), (W1′) and (W2′). Then the functional defined by
is well defined and of class and
Furthermore, the critical points of f in E are solutions of (1.1) with .
Proof Lemma 2.3 implies that f defined by (2.16) is well defined on E. Next, we prove that (2.17) holds. By (W2′), one can choose an such that
For any , there exists an integer such that for . Then, for any sequence with for and any number , by (W2′), (2.3) and Lemma 2.2, we have
where . Then by (2.16), (2.19) and Lebesgue’s dominated convergence theorem, we have
This shows that (2.17) holds. Observe that for ,
It follows from (2.17) and (2.20) that for all if and only if
So, the critical points of f in E are the solutions of system (1.1) with .
Let us prove now that is continuous. Let in E. Then there exists a constant such that
It follows from (2.1) that
By (W2′), one can choose an such that
From (2.3), (2.17), (2.21), (2.22), (2.23), (W2′) and the Hölder inequality, we have
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
Proof of Theorem 1.1 In view of Lemma 2.4, . In what follows, we first show that f is bounded from below. Choose , it follows from (2.12) that
By (2.16) and (3.1), we have
Since , (3.2) implies that as . Consequently, f is bounded from below.
Next, we prove that f satisfies the (PS)-condition. Assume that is a sequence such that is bounded and as . Then by (2.1) and (3.2), there exists a constant such that
So, passing to a subsequence if necessary, it can be assumed that in E. It is easy to verify that
Hence, we have by (3.3) and (3.4)
By virtue of (W2′), one can choose a such that
For any given number , we can choose an integer such that
It follows from (3.4) and the continuity of on x that there exists such that
On the other hand, it follows from (3.3), (3.5), (3.6), (3.7) and (W2′) that
Since ε is arbitrary, combining (3.8) with (3.9), we get
It follows from (2.17) that
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 .
Finally, we show that . Let and for . Then by (W1′), (W3′) and (2.16), we have
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. □
Proof of Theorem 1.2 In view of Lemma 2.4 and the proof of Theorem 1.1, is bounded from below and satisfies the (PS)-condition. It is obvious that f is even and . In order to apply Lemma 2.6, we prove now that there is a set such that K is homeomorphic to by an odd map, and . Let
where . Define
and
For any , there exist , such that
Then
and
where is a quadratic form. Since
and
Therefore, is a positive definite quadratic form. It follows that there exists an invertible matrix such that
Since all the norms of a finite dimensional normed space are equivalent, there is a constant such that
By (W1′), (W4′), (3.14), (3.16) and (3.19), we have
For , (3.20) implies that there exist and such that
Let
Then it follows from (3.17) that
By (3.18), we define a map as follows:
It is easy to verify that is an odd homeomorphic map. On the other hand, by (3.21), we have
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.
Example 4.1 In system (1.1), let be an real symmetric positive definite matrix for all , , and let
Then satisfies (L ν ) with , and
and
For any , there exist m integers , such that
Let and
Then
These show that all conditions of Theorem 1.2 are satisfied, where
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.
Example 4.2 In system (1.1), let be an real symmetric positive definite matrix for all , , and let
Then satisfies (L ν ) with , and
For any , there exist m integers , such that
Let and
Then
These show that all conditions of Theorem 1.2 are satisfied, where
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.
Example 4.3 In system (1.1), let be an real symmetric positive definite matrix for all , , and let
Then satisfies (L ν ) with , and
For any , there exist m integers , such that
Let and
Then
These show that all conditions of Theorem 1.2 are satisfied, where
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.
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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).
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XL wrote the first draft and XT corrected and improved the final version. Both authors read and approved the final draft.
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Lin, X., Tang, X. Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential. Adv Differ Equ 2013, 154 (2013). https://doi.org/10.1186/1687-1847-2013-154
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DOI: https://doi.org/10.1186/1687-1847-2013-154
Keywords
- homoclinic solution
- discrete Hamiltonian system
- critical point
- Clark’s theorem