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

摘要

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