Existence and nonexistence of solutions for singular quadratic quasilinear equations

TL;DR

This paper investigates the conditions under which solutions exist or do not exist for a class of nonlinear elliptic equations with singular gradient terms, establishing that the parameter b3<2 is critical for solution existence.

Contribution

It provides a precise characterization of the parameter b3 for the existence of solutions to singular quadratic quasilinear equations, extending understanding of such nonlinear problems.

Findings

Solutions exist if and only if b3<2 for sufficiently regular positive data.

The critical threshold b3=2 determines the existence or nonexistence of solutions.

The results apply to equations with natural growth in the gradient and singular lower order terms.

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.