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.
L. Damascelli. B. Sciunzi. "Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications." Differential Integral Equations 23 (5/6) 419 - 434, May/June 2010. https://doi.org/10.57262/die/1356019303