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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.

In this paper we prove monotonicity and symmetry properties of positive solutions of the equation $ - div (|Du|^{p-2}Du)=f(u)$, $1 0 $, $ c \geq 0 $, $ q \geq p-1 $. As a consequence we get an extension to the $p$--Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.