Quasilinear elliptic equations with singular quadratic growth terms

Abstract

In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{(1-u)^\gamma}=g & \mbox{in $\Omega$,}\newline \hfill u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} $$ where $\Omega $ is a bounded open set of $\mathbb{R}^N$, $g\geq 0 $ is a function in some Lebesgue space, and $\gamma>0$. We prove both existence and nonexistence of solutions depending on the value of $\gamma$ and on the size of $g$.