Maximum principle and existence of positive solutions for nonlinear systems on ℝN

Abstract

In this paper, we study the following non-linear system on ℝN -∆pu = α(x)|u|p-2 u + b(x)|u|α|v|βv + ƒ x ∈ ℝN -∆qv = c(x)|u|α|v|βu + d(x)|v|q-2 v + g x ∈ ℝN lim|x|→∞ u(x) = lim v(x) = 0, u, v > 0 in ℝN where ∆pu = div |∇u|p-2 ∇u) with p > 1 and p ≠ 2 is the "p-Laplacian", α, β > 0, p, q > 1, and ƒ, g are given functions. We obtain necessary and sufficient conditions for having a maximum principle; then we use an approximation method to prove the existence of positive solution for this system.