Weak Solutions for Nonlinear Neumann Boundary Value Problems with p(x)-Laplacian Operators

We study the nonlinear Neumann boundary value problem with a p(x)Laplacian operator {Delta(p(x))u + a(x) vertical bar u vertical bar(p(x)-2)u - f(x, u) in Omega, vertical bar del u vertical bar(p(x)-2)partial derivative u/partial derivative v - vertical bar u vertical bar(q(x)-2)u + lambda vertical bar u vertical bar(w(x)-2)u on partial derivative Omega, where Omega subset of R-N, with N >= 2, is a bounded domain with smooth boundary and q (x) is critical in the context of variable exponent p(*)(x) = (N - 1) p(x) / (N - p (x)). Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.