Multiple positive solutions of second-order nonlinear difference equations with discrete singular φ-Laplacian. Issue 1 (2nd January 2019) - Chen, Tianlan; Ma, Ruyun; Liang, Yongwen
ABSTRACT: We consider the discrete boundary value problems with mean curvature operator in the Minkowski space Δ Δ u k − 1 1 − ( Δ u k − 1 ) 2 + λ μ k ( u k ) q = 0, k ∈ [ 2, n − 1 ] Z, Δ u 1 = 0 = u n, whereλ > 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 ofC 1 -functionals, we show that there existsΛ > 0 such that the above problem has zero, at least one or two positive solutions according toλ ∈ ( 0, Λ ), λ = Λ orλ > Λ . Moreover, Λ is strictly decreasing with respect to n .