Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

Abstract

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is $$(P) \qquad \displaystyle{\left\{\begin{array}{ll} - \bigtriangleup _p u -\hbox{div}(c(x)|u|^{\gamma})+b(x)|\nabla u|^{\lambda} =\mu & \hbox{in}\; \Omega,\\ u=0 & \hbox{on}\; \partial\Omega, \end{array} \right.}$$ where \(\Omega \) is a bounded open subset of \(\mathbb{R}^{N} \), \(N\geq 2\), \(\bigtriangleup _p\) is the so-called \(p-\)Laplace operator, \(1< p< N\), \(\mu\) is a Radon measure with bounded variation on \(\Omega \), \(0\le\gamma\le p-1\), \(0\le\lambda\le p-1\), \(|c|\) and \(b\) belong to the Lorentz spaces \(L^{\frac{N}{p-1},\,r}(\Omega) \), \(\frac{N}{p-1}\leq r \leq +\infty\) and \(L^{N,1}(\Omega)\), respectively. In particular we prove the existence result under the assumption that \(\gamma=\lambda=p-1\), \(\|b\|_{L^{N,1}(\Omega)}\) is small enough and \({|c|\in L^{\frac{N}{p-1},r}(\Omega)}\), with \(r<+\infty\). We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prototype is \((P)\) with \(b\equiv 0\).