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We consider weak distributional solutions to the equation $-\Delta_pu=f(u)$in half-spaces under zero Dirichlet boundary condition. We assume that thenonlinearity is positive and superlinear at zero. For $p>2$ (the case$1<p\leq2$ is already known) we prove that any positive solution is strictlymonotone increasing in the direction orthogonal to the boundary of thehalf-space. As a consequence we deduce some Liouville type theorems for theLane-Emden type equation. Furthermore any nonnegative solution turns out to be$C^{2,\alpha}$ smooth.

Monotonicity in half-spaces of positive solutions to $-\Delta_p u=f(u)$ in the case $p>2$