Existence and Multiplicity of Solutions for a Robin Problem Involving the p(x)-Laplace Operator

We study the following nonlinear Robin boundary-value problem − Δ p ( x )u = λ f ( x , u ) in Ω ,| ∇ u |p ( x ) - 2 (∂ u / ∂ v) + β ( x )| u | p ( x ) − 2 u = 0 on ∂ Ω, where Ω ⊂ ℝ N is a bounded domain with smooth boundary ∂ Ω , ∂ u / ∂ v is the outer unit normal derivative on ∂ Ω , λ > 0 is a real number, p is a continuous function onΩ ¯with inf x ∈ Ω ¯p ( x ) > 1 , β ∈ L ∞( ∂ Ω ) with β − := inf x ∈ ∂Ω β ( x ) > 0 , and f : Ω × ℝ → ℝ is a continuous function. Using the variational method, under appropriate assumptions on f , we obtain results on existence and multiplicity of solutions.