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

We consider the discrete boundary value problems with mean curvature operator in the Minkowski space Delta[Delta u(k-1)/root 1 - (Delta u(k-1))(2)] + lambda mu(k)(u(k))(q) = 0, k is an element of [2, n - 1](Z), Delta u(1) = 0 = u(n), where lambda > 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 Lambda > 0 such that the above problem has zero, at least one or two positive solutions according to lambda is an element of(0, Lambda), lambda = Lambda, or lambda > Lambda. Moreover, Lambda is strictly decreasing with respect to n.