Existence of homoclinic solutions for a class of difference systems involving p -Laplacian
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RESEARCH
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Existence of homoclinic solutions for a class of difference systems involving p-Laplacian Qiongfen Zhang* * Correspondence: [email protected] College of Science, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China
Abstract By using the critical point theory, some existence criteria are established which guarantee that the difference p-Laplacian systems of the form (|u(n – 1)|p–2 u(n – 1)) – a(n)|u(n)|q–p u(n) + ∇W(n, u(n)) = 0 have at least one or infinitely many homoclinic solutions, where 1 < p < (q + 2)/2, q > 2, n ∈ Z, u ∈ RN , a : Z → (0, +∞), and W : Z × RN → R are not periodic in n. MSC: 34C37; 35A15; 37J45; 47J30 Keywords: homoclinic solutions; variational methods; difference p-Laplacian systems
1 Introduction Consider homoclinic solutions of the following p-Laplacian system: q–p p– u(n – ) u(n – ) – a(n)u(n) u(n) + ∇W n, u(n) = ,
n ∈ Z,
(.)
where < p < (q + )/, q > , n ∈ Z, u ∈ RN , a : Z → (, +∞), and W : Z × RN → R are not periodic in n. is the forward difference operator defined by u(n) = u(n + ) – u(n), u(n) = (u(n)). As usual, we say that a solution u of (.) is homoclinic (to ) if u(n) → as n → ±∞. In addition, if u(n) ≡ , then u(n) is called a nontrivial homoclinic solution. We may think of (.) being a discrete analogue of the following differential system: p– q–p d ˙ u(t) ˙ u(t) – a(t)u(t) u(t) + ∇W t, u(t) = , dt
t ∈ R.
(.)
When p = , (.) can be regarded as a discrete analogue of the following second-order Hamiltonian system: q– ¨ – a(t)u(t) u(t) + ∇W t, u(t) = , u(t)
t ∈ R.
(.)
Problem (.) has been studied by Shi et al. in [] and problem (.) has been studied in [–]. 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 ©2014 Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhang Advances in Difference Equations 2014, 2014:291 http://www.advancesindifferenceequations.com/content/2014/1/291
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chaotic phenomenon. Therefore, it is of practical importance to investigate the existence of homoclinic orbits of (.) emanating from . By applying critical point theory, the authors [–] 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 differenti
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