Renormalized solutions to elliptic equations with measure data in unbounded domains

Abstract

We consider the nonlinear elliptic problem − div ( a ( x , ∇ u ) ) + a 0 ( x , u ) = μ , in Ω , u = 0 on ∂ Ω , where Ω is an open (possibly unbounded) subset of R N , N ≥ 2 , μ is a Radon measure with bounded variation in Ω , and ( u ) ↦ − div ( a ( x , ∇ u ) ) + a 0 ( x , u ) is a monotone operator acting in W 0 1 , p ( Ω ) , 1 < p ≤ N . We prove that for every μ there exists at least a renormalized solution u to the problem, that is a distributional solution with additional summability properties. Moreover, if the operator is strictly monotone and μ does not charge sets of capacity zero, such a solution is unique.