  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 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.
