Quasilinear elliptic equations with singular quadratic growth terms

TL;DR

This paper investigates the existence and nonexistence of positive solutions for a class of singular quasilinear elliptic equations with quadratic gradient growth, depending on the parameter b3 and the data function g.

Contribution

It provides new results characterizing when solutions exist or do not exist for these singular equations based on b3 and g.

Findings

Existence of solutions for certain b3 and g values.

Nonexistence results under different conditions.

Dependence of solutions on the size of g and the singularity parameter b3.

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 $g$.