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

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is − \Delta_p u − div(c(x)|u|^\gamma) + b(x)|\nabla u|^\lambda = μ in , u = 0 on \partial \Omega, where \Omega is a bounded open subset of R^N, N\ge 2, \Delta_p u is the so-called p-Laplace operator, 1 < p < N, μ is a Radon measure with bounded variation on , 0\ge \gamma\ge p − 1, 0 \ge \lambda \ge p − 1, |c| and b belong to the Lorentz spaces L (N/(p−1) , r), N/(p−1)\ge r \ge +\infty and L(N,1), respectively. In particular we prove the existence result under the assumptions that \gamma=\lambda= p − 1, \|b|\_L(N,1) is small enough and |c| belongs to L (N/(p−1) ,r), with r < +\infty. We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prototype is the previous Dirichlet problem with b=0.