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 {−div(|∇u|p(x)−2∇u)+|u|p(x)−2u=λf(x,u),x∈Ω,t∈R∂u∂v=0,x∈∂Ω,t∈R where Ω⊂RnΩ⊂Rn is a bounded domain with smooth boundary ∂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 Ω¯ with infx∈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.