Monotonicity in half-spaces of positive solutions to $-{\mathrm{\Delta }}_{p}u=f(u)$ in the case $p>2$

We consider weak distributional solutions to the equation $-{\mathrm{\Delta }}_{p}u=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\le 2$ is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville type theorems for the Lane-Emden type equation. Furthermore any nonnegative solution turns out to be $C^{2,\alpha }$ smooth.