Boundary estimates for singular elliptic problems involving a gradient term

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 $\vartheta=0$.