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Regularity for the fractional Gelfand problem up to dimension 7

We study the problem $(-Δ)^su=λe^u$ in a bounded domain $Ω\subset\mathbb R^n$, where $λ$ 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 $n\leq7$ for all $s\in(0,1)$ whenever $Ω$ 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 $s\gtrsim0'28206...$, or if $n=9$ and $s\gtrsim0'63237...$. These results are new even in the unit ball $Ω=B_1$.