Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth

Abstract

We deal with the following nonlinear elliptic problem: $$ \begin{cases} - \hbox{div} \ {{\bf a}}( x , u, \nabla u) + b( x , u , \nabla u) = f & \ \hbox{ in } \ \Omega \\ u = 0 & \ \hbox{ on} \ \partial \Omega, \end{cases} $$ where $\Omega$ is a bounded, open set in $\mathbb R^N$, $f \in L^1 ( \Omega )$, $- \hbox{div} \ {{\bf a}}(x, u, \nabla u) $ defines an operator satisfying Leray---Lions-type conditions, and the lower-order term satisfies natural growth conditions and some other properties; we point out that these properties do not include a sign assumption (see in (2) below our model example). We prove existence of an entropy solution for this problem (see Definition 2.2 below), and we show that, under a natural monotonicity hypothesis, there exists a smallest entropy solution.