Existence of weak solution for nonlinear elliptic system

Abstract

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system

-div A(x,∇u) = -a(x)|u|^p(x)-2u - b(x)|u|^α(x)|v|^β(x)v + f(x) in Ω,

-div B(x,∇v) =  -c(x)|v|^q(x)-2v - d(x)|v|^β(x)|u|^α(x)u + g(x) in Ω,

u = v = 0 on ∂Ω,

where Ω is an open bounded domains of R^N with a smooth boundary ∂Ω. The existence of weak solutions is proved using the theory of monotone operators.

2000 Mathematics Subject Classification. Primary 35B45; Secondary 35J55.

Key words and phrases. Weak solutions; nonlinear elliptic systems; p(x)-Laplacian; monotone operators; generalized Lebesgue-Sobolev spaces.