Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

We prove a weak comparison principle in narrow unbounded domains for solutions to $-{\mathrm{\Delta }}_{p}u=f(u)$ in the case $2<p<3$ and $f(\cdot )$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot )$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.