Nonhomogeneous<i>p</i>(<i>x</i>)-Laplacian Steklov problem with weights

Abstract

This paper is concerned with a weighted Steklov problem involving the p(x)-Laplacian operator in Sobolev spaces with variable exponents −div(ξ(x)|∇u|p(x)−2∇u)+a(x)|u|p(x)−2u=λ∂F∂u(x,u),x∈Ω,ξ(x)|∇u|p(x)−2∂u∂n=β∂G∂u(x,u),x∈∂Ω. Our approach is based on variational method and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.