International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 907453, 21 pages
doi:10.1155/2010/907453

Existence of multiple solutions of a second-order difference boundary value problem

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.