  In this paper we consider classical solutions $u$ of the semilinear
fractional problem $(-\Delta)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in
$\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-\Delta)^s$, $0<s<1$, stands
for the fractional laplacian, $N\ge 2$, $\mathbb{R}^N_+=\{x=(x',x_N)\in
\mathbb{R}^N:\ x_N>0\}$ is the half-space and $f\in C^1$ is a given function.
With no additional restriction on the function $f$, we show that bounded,
nonnegative, nontrivial classical solutions are indeed positive in
$\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in
} \mathbb{R}^N_+. $$ This is in contrast with previously known results for the
local case $s=1$, where nonnegative solutions which are not positive do exist
and the monotonicity property above is not known to hold in general even for
positive solutions when $f(0)<0$.
