Existence and asymptotic behavior of positive solutions for a class of (p(x),q(x))- Laplacian systems

Abstract

In this paper, our main purpose is to establish the existence of positive solution of the following system ⎧ ⎪ ⎪ −Δp(x)u = u α(x) + λ p(x) v m(x) , x ∈ Ω −Δq(x)v = v β(x) + θ q(x) u n(x) , x ∈ Ω u = v = 0, x ∈ ∂ Ω, where Ω ⊂ R N is a bounded domain with C 2 boundary, p(x),q(x) are functions which satisfy some conditions, −Δp(x)u = −div(|∇u| p(x)−2 ∇u) is called p(x)-Laplacian. We give the exis- tence results of positive solutions and consider the asymptotic behavior of the solutions near the boundary. The approach is based on the sub- and super-solution method.