Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)su=λeu in a bounded domain Ω⊂Rn, 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 xi-direction and symmetric with respect to {xi=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=B1.