Existence and Nonexistence of a Solution for a Nonlinear p(x)-Elliptic Problem with Right-Hand Side Measure

We discuss the existence and nonexistence of solution of a nonlinear problem p(x)-elliptic-div(a(x,∇u))+g(x,u,∇u)=μ, where μ is a Radon measure with bounded total variation, by considering the Sobolev spaces with variable exponents. This study is done in two cases: (i) μ is absolutely continuous with respect to p(x)-capacity. and (ii) μ is concentrated on a Borel set of null p(x)-capacity