Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis

2017/04/09 by Barrios, B., Del Pezzo, L., García-Melián, J. +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.02597

Abstract

In this paper we analyze the semi-linear fractional Laplace equation (-Δ)^s u = f(u) in ℝ^N₊, u=0 in ℝ^N∖ ℝ^N₊, where ℝ^N₊=\x=(x',x_N)∈ ℝ^N: x_N>0\ stands for the half-space and f is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if u is a bounded solution with ρ:=sup_ℝ^Nu verifying f(ρ)=0, then u is necessarily one-dimensional.

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• Boundary regularity and a priori estimates for fractional equations on unbounded domains

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