Regularity for the fractional Gelfand problem up to dimension 7

Abstract

We study the problem $(-\Delta {)}^{s}u=\lambda e^u$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$, where $\lambda$ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions $n\le 7$ for all $s\in (0,1)$ whenever $\Omega$ is, for every $i=1,...,n$, convex in the $x_i$-direction and symmetric with respect to $\{x_i=0\}$. The same holds if $n=8$ and $s\gtrsim 0^{\prime }\mathrm{28206...}$, or if $n=9$ and $s\gtrsim 0^{\prime }\mathrm{63237...}$. These results are new even in the unit ball $\Omega =B_1$.