Exact solutions of nonlinear conformally invariant integral equations in R3

In this paper, we will study the entire positive C-4 solutions of the geometrically interesting integral equation: u(x) = 1/8 pi integral(3)(R) vertical bar x -y vertical bar u(-q)(y)dy with 0 < q in R-3. We will show that there are positive entire solutions which take the form u(x) = c(1+vertical bar x vertical bar(2))(1/2) up to dilation and translations, only when q = 7. (c) 2004 Elsevier Inc. All rights reserved.