Existence and nonexistence of solutions for singular quadratic quasilinear equations

Abstract

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{u^{\gamma}} = f & \mbox{in } \Omega,\newline \hfill u=0 \hfill & \mbox{on } \partial \Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N $, $\gamma> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $\Omega$. As a consequence of our main results, we prove that the condition $\gamma<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(\Omega)$ for every sufficiently regular $f$ as above.