Variational Approach for a Robin Problem Involving Non Standard Growth Conditions

Abstract

In this manuscript, we study a Robin problem driven by the p(x)-Laplacian with two parameters. −div(|∇w|p(x)−2∇w)=λV(x)|w|q(x)−2w,x∈Q, |∇w|p(x)−2∂w∂n+θ(x)|w|p(x)−2=βV1(x)|w|r(x)−2w.x∈∂Q. Here, Q is a regular bounded domain in RN, λ,β>0, p,q are continuous functions on Q¯, ∂w∂n is the outer unit normal derivative on ∂Q, θ∈L∞(∂Q), such that essinfx∈∂Qθ(x)>0,V is an indefinite function in Ls(x)(Q) and V1 is a non-negative one in Ls1(x)(∂Q). Using variational tools, we show the existence of a non-trivial solution.