Achievability of a supremum for the Hardy-Littlewood-Sobolev inequality with supercritical exponent

In this paper, we prove that the supremum sup {integral(B)integral(B) vertical bar u(y)vertical bar(p(vertical bar y vertical bar))vertical bar u(x)vertical bar(p(vertical bar x vertical bar)) /vertical bar x - y vertical bar(mu) dxdy : u is an element of H-0,H- (1)(rad) (B), parallel to del u parallel to(L2(B)) = 1} is attained, where B denotes the unit ball in R-N (N >= 3), mu is an element of (0,N), p(r) = 2(mu)* + r(t), t is an element of (0, min{N/2 - mu/4, N - 2}) and 2(mu)* = (2N - mu)/(N - 2) is the critical exponent for the Hardy-Littlewood-Sobolev inequality.