Homoclinic solutions of discrete nonlinear Schrödinger equations with unbounded potentials

Abstract

In this paper, the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations −Δuk+vkuk−ωuk=fk(uk)are considered. With asymptotical nonlinearity and unbounded potential V={vk:k∈Z} satisfying lim∣k∣→∞vk=∞, we achieve the existence of nontrivial homoclinic solution via the saddle point reduction method and Conley index theory. Furthermore, we can prove there are at least two nontrivial homoclinic solutions with extra assumptions about the nonlinearity fk.