Robin problems involving the p(x)-Laplacian

By applying Mountain Pass Lemma and Ekeland's variational principle, we prove two different situations of the existence of solutions for the following Robin problem −Δp(x)u=λV(x)|u|q(x)−2uinΩ,|∇u|p(x)−2∂u∂ν+β(x)|u|p(x)−2u=0on∂Ω,where Ω⊂RN (N ≥ 2) is a bounded smooth domain, V is an indefinite weight function which can change sign in Ω and p,q:Ω¯→(1,+∞) are continuous functions.