- Research Article
- Open Access
© Alberto Cabada et al. 2010
- Received: 5 July 2010
- Accepted: 27 October 2010
- Published: 31 October 2010
- Banach Space
- Hamiltonian System
- Difference Equation
- Laplacian Equation
- Variational Approach
In the theory of differential equations, a trajectory , which is asymptotic to a constant as is called doubly asymptotic or homoclinic orbit. The notion of homoclinic orbit is introduced by Poincaré  for continuous Hamiltonian systems.
In the recent paper of Li  a unified approach to the existence of homoclinic orbits for some classes of ODE's with periodic potentials is presented. It is based on the Brezis and Nirenberg's mountain-pass theorem . In this paper we extend this approach to homoclinic orbits for discrete -Laplacian type equations.
Discrete boundary value problems have been intensively studied in the last decade. The studies of such kind of problems can be placed at the interface of certain mathematical fields, such as nonlinear differential equations and numerical analysis. On the other hand, they are strongly motivated by their applicability to mathematical physics and biology.
The variational approach to the study of various problems for difference equations has been recently applied in, among others, the papers of Agarwal et al. , Cabada et al. , Chen and Fang , Fang and Zhao , Jiang and Zhou , Ma and Guo , Mihăilescu et al. , Kristály et al. .
satisfies the Rabinowitz's type condition:
In order to obtain homoclinic solutions of (1.10), we will use variational approach and Brezis-Nirenberg mountain pass theorem .
Our main result is the following.
The paper is organized as follows. In Section 2, we present the proof of the main result and discuss the optimality of the condition . In Section 3, we give some examples of equations modeled by this kind of problems and present some additional remarks.
We have the following result.
are continuous functions.
Theorem 2.2 (mountain-pass theorem, Brezis and Nirenberg ).
Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Now, let be a nonzero homoclinic solution of problem (1.10). Assume that it attains positive local maximums and/or negative local minimums at infinitely many points . In particular we can assume that . In consequence and .
Suppose now that function vanishes at infinitely many points . From condition we conclude that and, in consequence, . Therefore it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion.
To illustrate the optimality of the obtained results, we present in the sequel an example in which it is pointed out that condition cannot be removed to deduce the existence result proved in Theorem 1.1.
arising in mathematical physics and biology.
(B) Higher Even-Order Difference Equations.
Now following the steps of the proof of Theorem 1.1 one can prove the following.
Suppose that , the function is positive and -periodic and the functions satisfy assumptions − and , . Then, for each , (3.8) has a nonzero homoclinic solution , which is a critical point of the functional .
A typical example of (3.8) is (3.2), which is a discretization of a fourth-order extended Fisher-Kolmogorov equation. Homoclinic solutions for fourth-order ODEs are studied in  using variational approach and concentration-compactness arguments. As a consequence of Theorem 3.2 we obtain the following corollary.
(C) Second-Order Difference Equations with Cubic and Quintic Nonlinearities.
Our next example is (3.3), known as stationary Ginzubrg-Landau equation with cubic-quintic nonlinearity. We refer to [18, 19] and references therein. From physical point of view it is interesting the case , , . Theorem 1.1 can be applied for with , -periodic, and positive. Then satisfies assumptions − with and as a consequence we have the following corollary.
the homoclinic solution of (1.10) is positive.
We summarize above observations in the following.
Suppose that the function is positive and -periodic and the functions satisfy assumptions , , and . Then, for each , (1.10) has a nonzero homoclinic solution . If moreover holds, is a positive solution on that is strictly monotone for large enough.
We summarize above observation in the following.
satisfies the estimate (3.15).
Clearly, the positive solution of (3.17) is a positive solution of (3.4) too.
We summarize above facts in the following.
This work is dedicated to Professor Gheorghe Moroşanu on the occasion of his 60-th birthday.
S. Tersian is thankful to Department of Mathematical Analysis at University of Santiago de Compostela, Spain, where a part of this work was prepared during his visit. A. Cabada partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2007-61724.
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