A blow-up approach for singular elliptic problems with natural growth in the gradient

TL;DR

This paper establishes existence and nonexistence results for a class of singular elliptic problems with gradient-dependent terms, using a blow-up approach and fixed point theory to handle singularities and superlinear growth.

Contribution

It introduces a novel blow-up method combined with fixed point theory to analyze elliptic problems with singular gradient terms and superlinear zero order terms.

Findings

Existence results for certain parameter ranges.

Nonexistence results under specific conditions.

Development of a blow-up technique for singular problems.

Abstract

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in \Omega, \\ u=0, &x\in\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N (N\geq 3)$ is a bounded smooth domain, $\lambda>0$, $p>1$, $\delta\geq 0$, $\gamma>0$ and $\mu\in L^\infty(\Omega)$. The main achievement resides in handling a possibly singular ($\delta=0$) first order term having a nonconstant coefficient $\mu$ in the presence of a superlinear zero order term. Our approach for the existence results is based on fixed point theory. With the aim of applying it, a previous analysis on a related non-homogeneous problem is carried out. The required a priori estimates are proven via a blow-up method.