Discrete Neumann boundary value problem for a nonlinear equation with singular ϕ-Laplacian

Abstract

Let I ⊂ R be an open interval with 0 ∈ I, and let g ∈ C 1 (I, (0, +∞)). Let N ∈ N be an integer with N ≥ 4, [2, N-1] Z := {2, 3,. .. , N-1}. We are concerned with the existence of solutions for the discrete Neumann problem ⎧ ⎨ ⎩ ∇(k n-1 v k √ 1-(v k) 2) = nk n-1 [-g (ψ-1 (v k)) √ 1-(v k) 2 + g(ψ-1 (v k))H(ψ-1 (v k), k)], k ∈ [2, N-1] Z , v 1 = 0 = v N-1 which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where ψ(s) := s 0 dt g(t) , ψ-1 is the inverse function of ψ, and H : R × [2, N-1] Z → R is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.