Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (-Delta)(s)u = lambda e(u) in a bounded domain Omega subset of 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 <= 7 for all s is an element of (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 greater than or similar to 0.28206 ... , or if n = 9 and s greater than or similar to 0.63237... These results are new even in the unit ball Omega = B-1. (C) 2014 Elsevier Inc. All rights reserved.