Nonlinear elliptic equations on S-n in the supercritical case

We study some elliptic nonlinear differential equations on S-n, when the eq uations are invariant under the group G = O(m) x O(k) subset of O(n + 1) wi th m + k = n + and k greater than or equal to m greater than or equal to 2. We find the best constants in the Sobolev inequalities which deal with the supercritical case. We apply these results to solve the equations (1) Delt a phi + a(x)phi = f(x)phi(q-1) when n. 1 4 and q = 2k/ (k - 2) and (2) Delt a phi + a(x) = f(x)e(phi) when n = 3. (C) Elsevier, Paris.