Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution

We consider the class of semi-stable positive solutions to semilinear equations - ?u = f(u) in a bounded domain O ? R n of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n - m variables. We assume 2 = m = n - 2. When the domain is convex, we establish a priori L p and bounds for each dimension n, with p = 8 when n = 7. These estimates lead to the boundedness of the extremal solution of - ?u = ?f(u) in every convex domain of double revolution when n = 7. The boundedness of extremal solutions is known when n = 4 for any domain O, and in dimensions 5 = n = 9 in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities f in dimensions 5 = n = 9.