Copyright © 2010 Bo Zheng and Huafeng Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ2u(n−1)+V′(u(n))=0, n∈ℤ(1,T), u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue λk  (k≥2) of linear difference problem Δ2u(n−1)+λu(n)=0, n∈ℤ(1,T), u(0)=0=u(T+1) near infinity and the trivial solution of the first equation is a local minimizer under some assumptions on V.