Existence and multiplicity results for the p(x)–Laplacian system

This paper deals with the existence and multiplicity of nontrivial weak solutions for the following system involving variable exponents{█(-Δ_p(x) u=λf(v), in Ω,@-Δ_p(x) v=λg(u), in Ω,@u=v=۰ on ∂Ω,)┤where Ω=B(۰,R) is a bounded domain of R^N, λ is a positive real parameter and p is real continuos function on Ω ̅. Using a variational method and Krasnoselskii,s genus theory, we would show the existence and multiplicity of the solutions. For this purpose, we focus on a generalized variable exponent Lebesgue-Sobolev space.

Keywords:

p(x)–Laplacian , variational method , genus theory , Sobolev space.