Existence and nonexistence of a solution for a nonlinear \(p(x)\)-elliptic problem with right-hand side measure (Q1637826)

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