Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System

Abstract

This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, S μ 1 , μ 2 q 1 , q 2 − div q 1 x ∇ u p − 2 ∇ u = μ 1 u p − 2 u + α + 1 u α − 1 u v β + 1 in Ω − div q 2 x ∇ v p − 2 ∇ v = μ 2 v p − 2 v + β + 1 u α + 1 v β − 1 v in Ω u > 0 , v > 0 in Ω u = v = 0 on ∂ Ω , where Ω ⊂ ℜ N N ≥ 3 , 2 ≤ p < N , q 1 , q 2 ∈ W 1 , p Ω ∩ C Ω ¯ , α , β ≥ 0 , μ 1 , μ 2 , ≥ 0 and α , β > 0 satisfy α + β = p ∗ − 2 with p ∗ = p N / N − p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q 1 , q 2 near their minima and the dimension N . Moreover, we use the well known Pohozaev identity for prove the nonexistence result.