Existence of weak solution for nonlinear elliptic system
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 RN 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.