On the antimaximum principle for the discrete p-Laplacian with sign-changing weight

Abstract

This work deals with the antimaximum principle for the discrete Neumann and Dirichlet problem−Δφp(Δu(k−1))=λm(k)|u(k)|p−2u(k)+h(k)in[1,n].We prove the existence of three real numbers 0 ≤ a < b < c such that, if λ ∈ ]a, b[, every solution u of this problem is strictly positive (maximum principle), if λ ∈ ]b, c[, every solution u of this problem is strictly negative (antimaximum principle) and if λ=b, the problem has no solution. Moreover these three real numbers are optimal.