Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or L1 data

Abstract

We investigate solutions to nonlinear elliptic Dirichlet problems of the type{−divA(x,u,∇u)=μinΩ,u=0on∂Ω, where Ω is a bounded Lipschitz domain in Rn and A(x,z,ξ) is a Carathéodory function. The growth of the monotone vector field A with respect to the (z,ξ) variables is expressed through some N-functions B and P. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data, we prove existence and regularity of solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces. For L1-data problems we infer also uniqueness.