Solutions for Neumann boundary value problems involving p(x)p(x)-Laplace operators

In this paper we study the nonlinear Neumann boundary value problem of the following form: equation(P){−div(|∇u|p(x)−2∇u)+|u|p(x)−2u=λf(x,u)in Ω,|∇u|p(x)−2∂u∂ν=μg(x,u)on ∂Ω. Using the variational method, under appropriate assumptions on ff and gg, we obtain a number of results on existence and multiplicity of solutions. Also, what is also interesting is that we obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solution of the problem is considered.