Boundary estimates for singular elliptic problems involving a gradient term

TL;DR

This paper derives boundary behavior estimates and symmetry results for solutions to a class of singular quasilinear elliptic problems involving gradient terms, extending known results even in classical cases.

Contribution

It provides new boundary derivative estimates and symmetry results for solutions to singular elliptic problems with gradient terms, applicable even in classical settings.

Findings

Derived precise boundary derivative estimates for positive solutions.

Proved symmetry of solutions in symmetric convex domains.

Extended results to cases with gradient terms and singular nonlinearities.

Abstract

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and $f:[0,+\infty)\to\mathbb{R}$ is a locally Lipschitz continuous function. We obtain a precise estimate for directional derivatives of positive solutions in a neighborhood of the boundary. We also deduce the symmetry of positive solutions to the problem in a bounded symmetric convex domain. Our results are new even in the case $p=2$ and $\vartheta=0$.