Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

We study the regularity of stable solutions to the problem $$\{\begin{array}{rclllll}(-\mathrm{\Delta }{)}^{s}u& =& f(u)& \text{in}{B}_1,u& \equiv & 0& \text{in}{\mathbb{R}}^n\setminus B_1,\end{array}$$ where $s\in (0,1)$. Our main result establishes an $L^{\mathrm{\infty }}$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$\mathrm{.}Inparticular,thisestimateholdsforall$s\in(0,1)$indimensions$2 \leq n\leq 6$\mathrm{.}Itappliestoallnonlinearities$f\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of the extremal solution when $f$ is replaced by $\lambda f$ with $\lambda >0$. This is a widely studied question for $s=1$, which is still largely open in the nonradial case both for $s=1$ and $s<1$.