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

Abstract

We study the regularity of stable solutions to the problem $$\{\begin{array}{rcllccc}(-\Delta {)}^{s}u& =& f(u)& \text{in}{B}_1,u& \equiv & 0& \text{in}{\mathbb{R}}^n\setminus B_1,\end{array}$$ where $s\in (0,1)$. Our main result establishes an $L^{\infty }$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2\le n<2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in (0,1)$ in dimensions $2\le n\le 6$. It applies to all nonlinearities $f\in C^2$. 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 $s<1$.

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