Quasilinear elliptic problems with singular nonlinearities in half-spaces

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-{\mathrm{\Delta }}_{p}u=f(u)$ in ${\mathbb{R}}_{+}^N$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\mathrm{\infty })\to \mathbb{R}$ is a locally Lipschitz continuous function with a possible singularity at zero. Classification results for the case $f(u)=\frac{1}{u^{\gamma }}$ with $\gamma >0$ are also provided.