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

Abstract

In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta {)}^{s}u=f(u)\text{ in }{\mathbb{R}}_{+}^N,u=0\text{ in }{\mathbb{R}}^N\setminus {\mathbb{R}}_{+}^N,$$ where ${\mathbb{R}}_{+}^N=\{x=(x^{\prime },x_N)\in {\mathbb{R}}^N:x_N>0\}$ 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 $\rho :={\sup }_{{\mathbb{R}}^N}u$ verifying $f(\rho )=0$, then $u$ is necessarily one-dimensional.

30 pages, 1 figure