Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

This paper studies the existence of multiple solutions of the second-order difference boundary value problemΔ2u(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 eigenvalueλk (k≥2)of linear difference problemΔ2u(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 onV.