Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications

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.