Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data

摘要

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is (P) {-Delta(p) u - div(c(x)u(gamma)) + b(x)del u(lambda) = mu in Omega, u = 0 on partial derivative Omega, where Omega is a bounded open subset of R-N, N >= 2, Delta(p) is the so-called p-Laplace operator, 1 < p < N, mu is a Radon measure with bounded variation on Omega, 0 <= gamma <= p - 1, 0 <= lambda <= p - 1, |c| and b belong to the Lorentz spaces L N/p-1, r(Omega), N/p-1 <= r <= + infinity and L-N,L-1(Omega), respectively. In particular we prove the existence result under the assumption that gamma = lambda = p - 1, parallel to b parallel to(LN,1(Omega)) is small enough and |c| is an element of L N/p-1,(r)(Omega), with r < + infinity. We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prototype is (P) with b = 0.