Monotonicity and one-dimensional symmetry for solutions of −Δ p u = f(u) in half-spaces

Abstract

We prove a weak comparison principle in narrow domains for sub-super solutions to − p u = f (u) in the case 1 < p ≤ 2 and f locally Lipschitz continuous. We exploit it to get the monotonicity of positive solutions to − p u = f (u) in half spaces, in the case 2N +2 N +2 < p ≤ 2 and f positive. Also we use the monotonicity result to deduce some Liouville-type theorems. We then consider a class of sign-changing nonlinearities and prove a monotonicity and a one-dimensional symmetry result, via the same techniques and some general a-priori estimates. Mathematics Subject Classification (2000)

On the Höpf boundary lemma for quasilinear problems involving singular nonlinearities and applications

“…where ϑ := σC 2 C 1 > 0 is sufficiently small when σ > 0 is sufficiently small. Applying again Lemma 2.1 of [15] it follows that C(R) (|∇u| + |∇u τ |) p−2 |∇w τ,ε | 2 dx = 0 for any R ≥ R 0 . This provides a contradiction exactly as in the case 1 < p < 2 so that the thesis follows also in the case p ≥ 2.…”

“…with σ := (α − γ)β + 1. We take α > 0 sufficiently large so that σ > 0 and ϑC 2 αC 1 < 2 −σ so that Lemma 2.1 of [15] apply and shows that L(R) = 0. ¿From this it follows that u ≤ v + ε for every ε > 0, hence u ≤ v. Arguing in the same way it follows that u ≥ v and this proves the uniqueness result.…”

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

“…2 Remark 16. Ifλ is given by (17) we are in the hypothesis of Lemma 15 for any x 0 ∈ R, since the difference wλ = u − uλ fulfills an equation of the type (12) and we can argue as in (16) to prove that, ∂u ∂ y (x 0 ,λ) > 0.…”

“…We believe that the techniques developed in [17] might also be useful in the fully nonlinear case, but this would in any case require u or |∇u| be bounded.…”

“…In a recent paper [17], the authors prove a monotonicity result for positive weak solutions to p u + f (u) = 0 in n-dimensional half spaces. We believe that the techniques developed in [17] might also be useful in the fully nonlinear case, but this would in any case require u or |∇u| be bounded.…”