MONOTONICITY OF THE SOLUTIONS OF SOMEQUASILINEAR ELLIPTIC EQUATIONS IN THEHALF-PLANE, AND APPLICATIONS

摘要

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