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

Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results