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In this paper we analyze the semi-linear fractional Laplace equation$$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in} \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in\mathbb{R}^N:\ x_N>0\}$ stands for the half-space and $f$ is a locallyLipschitz nonlinearity. We completely characterize one-dimensional boundedsolutions of this problem, and we prove among other things that if $u$ is abounded solution with $\rho:=\sup_{\mathbb{R}^N}u$ verifying $f(\rho)=0$, then$u$ is necessarily one-dimensional.

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