  We study stable solutions to fractional semilinear equations $(-\Delta)^s u =
f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and
under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$
with general $g$. We establish a uniqueness and a classification result, and we
show that weak (energy) stable solutions can be approximated by a sequence of
bounded (and hence regular) stable solutions to similar problems.
  As an application of our results, we establish the interior regularity of
weak (energy) stable solutions to the problem for the half-Laplacian in
dimensions $1 \leq n \leq 4$.
