Existence and multiplicity of solutions for a Neumann problem involving the p(x)p(x)-Laplace operator

In this paper, we study the following nonlinear Neumann boundary value problem View the MathML source{−div(|∇u|p(x)−2∇u)+|u|p(x)−2u=λf(x,u),x∈Ω,t∈R∂u∂v=0,x∈∂Ω,t∈R Turn MathJax on where Ω⊂RnΩ⊂Rn is a bounded domain with smooth boundary View the MathML source∂Ω,∂u∂v is the outer unit normal derivative on ∂Ω∂Ω, λ>0λ>0 is a real number, pp is a continuous function on View the MathML sourceΩ¯ with View the MathML sourceinfx∈Ω¯p(x)>1,f:Ω×R→R is a continuous function. Using the three critical point theorem due to Ricceri, under the appropriate assumptions on ff, we establish the existence of at least three solutions of this problem. Some known results are generalized.