Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Abstract

We consider the differential problem(*){A(u)=μinΩ,u=0on∂Ω,where Ω is a bounded, open subset of RN, N ≥ 2, A is a monotone operator acting on W01,p(Ω), p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p-capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L1(Ω) and of a measure in W−1,p′(Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*).