We study in this paper a nonlinear anisotropic problem with homogeneous Neumann boundary condition and Radon diffuse measure data which does not charge the sets of zero p(⋅)-capacity. We first prove, by using the techniques of monotone operators in Banach spaces, the existence of weak solutions and by approximation methods, the existence and uniqueness of entropy solution.

generalized Lebesgue-Sobolev spaces; anisotropic Sobolev spaces; weak solution; entropy solution; Neumann boundary condition; bounded Radon diffuse measure; Marcinkiewicz spaces