MULTIPLE SOLUTIONS FOR A DISCRETE NONLINEAR BOUNDARY VALUE PROBLEM VIA CRITICAL POINT THEORY

Abstract. The aim of my talk is to point out several results concerning the exis-tence and multiplicity of solutions for the following problem (P f,qλ) { −∆(φp(∆uk−1)) + qk|uk|p−2uk = λf(k, uk), k ∈ [1, N], u0 = uN+1 = 0, where, N is a fixed positive integer, [1, N] is the discrete interval {1,..., N}, f: [1, N]×R → R is a continuous function, ∆uk−1 = uk−uk−1 is the forward difference operator, qk ∈ R+0 for all k ∈ [1, N], φp(s): = |s|p−2s, 1 < p < +∞, and λ ∈ R+. The approach is based on recent critical point theorems. References [1] R. P. Agarwal, K. Perera and D. O’Regan, Multiple positive solutions of singular and nonsin-gular discrete problems via variational methods, Nonlinear Anal. 58 (2004), 69-73. [2] G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical points methods, Nonlinear Anal., to appear. [3] G. Bonanno and P. Candito, Non-differentiable functions with applications to elliptic equation