In this paper we study the existence and uniqueness of entropy solution to the class of nonlinear p(⋅)-Laplace problems of the type div(Φ(∇u−Θ(u)))+|u|p(x)−2u+α(u)∋μ with μ a diffuse measure and a Neumann nonhomogeneous boundary conditions of the form Φ(∇(u)−Θ(u))⋅η+β(u)=g. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

nonlinear elliptic; maximal monotone graph; Radon measure; entropy solution