Monotonicity in Half-Spaces of Positive Solutions to -Δpu = f (u) in the Case p > 2
Résumé
We consider weak distributional solutions to the equation -Δpu = f (u) in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For p > 2 (the case 1 < p < 2 is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville-type theorems for the Lane-Emden-type equation. Furthermore any nonnegative solution turns out to be C2'α smooth. \textcopyright 2017 Scuola Normale Superiore. All rights reserved.