Electron. J. Diff. Equ., Vol. 2015 (2015), No. 273, pp. 1-13.

Multiple homoclinic solutions for indefinite second-order discrete Hamilton system with small perturbation

Liang Zhang, Xianhua Tang

Abstract:
In this article, we sutdy the multiplicity of homoclinic solutions to the perturbed second-order discrete Hamiltonian system
$$ \Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\nabla W(n,u(n))+\theta\nabla F(n,u(n))=0, $$
where L(n) and W(n,x) are neither autonomous nor periodic in n. Under the assumption that W(n,x) is only locally superquardic as $|x|\to \infty$ and even in x and F(n,x) is a perturbation term, we establish some existence criteria to guarantee that the above system has multiple homoclinic solutions by minimax method in critical point theory.

Submitted June 9, 2015. Published October 21, 2015.
Math Subject Classifications: 39A11, 58E05, 70H05.
Key Words: Critical point; discrete Hamilton system; homoclinic solution; small perturbation

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Liang Zhang
School of Mathematical Sciences
University of Jinan
Jinan 250022, China
email: mathspaper2012@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha 410083, China
email: tangxh@csu.edu.cn

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