Multiple positive solutions of second-order nonlinear difference equations with discrete singular φ-Laplacian

ABSTRACTWe consider the discrete boundary value problems with mean curvature operator in the Minkowski space ΔΔuk−11−(Δuk−1)2+λμk(uk)q=0,k∈[2,n−1]Z,Δu1=0=un, where λ>0 is a parameter, n>4 and q>1. Using upper and lower solutions, topological methods and Szulkin's critical point theory for convex, lower semicontinuous perturbations of C1-functionals, we show that there exists Λ>0 such that the above problem has zero, at least one or two positive solutions according to λ∈(0,Λ), λ=Λ or λ>Λ. Moreover, Λ is strictly decreasing with respect to n.