  We investigate stable solutions of elliptic equations of the type
\begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad
{\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial
B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$,
$\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The
operator $(-\Delta)^s$ stands for the fractional Laplacian, a
pseudo-differential operator of order $2s$. According to the value of
$\lambda$, we study the existence and regularity of weak solutions $u$.
