Quasilinear elliptic problems with singular nonlinearities in half-spaces

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