  We consider weak distributional solutions to the equation $-\Delta_pu=f(u)$
in half-spaces under zero Dirichlet boundary condition. We assume that the
nonlinearity is positive and superlinear at zero. For $p>2$ (the case
$1<p\leq2$ 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.
