Boundary estimates for singular elliptic problems involving a gradient term

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-{\mathrm{\Delta }}_{p}u+\vartheta \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q=\frac{1}{u^{\gamma }}+f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma >0$, $0<q\le p$, $\vartheta \ge 0$ and $f:[0,+\mathrm{\infty })\to \mathbb{R}$ 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 $\vartheta =0$.