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

We study stable solutions to fractional semilinear equations $(-\mathrm{\Delta }{)}^{s}u=f(u)$ in $\mathrm{\Omega }\subset {\mathbb{R}}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in ${\mathbb{R}}^n\setminus \mathrm{\Omega }$ 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 $1\le n\le 4$.