Liouville results for m-Laplace equations in a half plane in R2

We consider weak positive solutions of the equation $-\Delta_m u=f(u)$ in the half-plane with zero Dirichlet boundary conditions. Assuming that the nonlinearity $f$ is locally Lipschitz continuous and $f(s)>0$ for $s>0$, we prove that any solution is monotone. Some Liouville type theorems follow in the case of Lane-Emden-Fowler type equations. Assuming also that $|\nabla u|$ is globally bounded, our result implies that solutions are one-dimensional, and the level sets are flat.