Quasilinear elliptic equations with natural growth

Abstract

In this paper we deal with the problem $$\left\{ \begin{array}{rcl} - {\rm div}\, (a(x,u)\nabla u) +{g(x,u,\nabla u)} & = & \lambda h(x)u + f{\mbox{ in }}\Omega,\\ u & = & 0{\mbox{ on }}\partial\Omega. \end{array} \right. $$ The main goal of the work is to get hypotheses on $a$, $g$ and $h$ such that the previous problem has a solution for all $\lambda>0$ and $f\in L^1(\Omega)$. In particular, we focus our attention in the model equation with $a(x,u)= (1+|u|^m)$, $g(x,u,\nabla u)=\frac{m}{2}|u|^{m-2}u|\nabla u|^2$ and $h(x)=\dfrac{1}{|x|^2}$.