Regularity for the fractional Gelfand problem up to dimension 7

We study the problem ( − Δ ) s u = λ e u in a bounded domain Ω ⊂ 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 ≤ 7 for all s ∈ ( 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 ≳ 0.28206 . . . , or if n = 9 and s ≳ 0.63237 . . . . These results are new even in the unit ball Ω = B 1 .