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

Abstract EN

We study the following nonlinear Robin boundary-value problem −Δp(x)u=λf(x,u) in Ω, |∇u|p(x)-2(∂u/∂v)+β(x)|u|p(x)−2u=0 on ∂Ω, where Ω⊂ℝN 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 on Ω¯ with infx∈Ω¯p(x)>1, β∈L∞(∂Ω) with β−:=infx∈∂Ωβ(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.