Monotonicity in Half-Spaces of Positive Solutions to -Δ_pu = f (u) in the case p > 2

Alberto Farina (1) , Luigi Montoro (2) , Berardino Sciunzi (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 Unical - Università della Calabria [Arcavacata di Rende]

We consider weak distributional solutions to the equation -Δ_pu = 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 C^2'α smooth.

Domaines: Mathématiques [math]

Alberto Farina, Luigi Montoro, Berardino Sciunzi. Monotonicity in Half-Spaces of Positive Solutions to -Δ_pu = f (u) in the case p > 2. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2017, 17 (4), pp.1207-1229. ⟨hal-03700891⟩