On the compactness problem of extremal functions to sharp Riemannian LP-Sobolev inequalities

Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n >= 2. For 1 <p <q(0) = min(2, root n), Djadli and Druet (2001) [13] proved the existence of extremal functions to the following sharp Riemannian LP-Sobolev inequality: parallel to u parallel to(p)(Lp*(M)) <= K(n,p)(p) parallel to del u parallel to(p)(pL(M)) + B-0(p,g)parallel to u parallel to(Lp(M))'(p) where p* = np/n-p and K (n, p)(p) and B-0(p, g) stands for, respectively, the first and second Sobolev best constants for this inequality. Let then E-g(p) be the corresponding extremal set normalized by the unity L-p*-norm. In contrast what happens in the whole space R" for 1 < p < n and in the Euclidean sphere S-n for p = 2, we establish the C-0-compactness of E-g(p) for any 1 < p < q0. Moreover, we address the question from a uniform viewpoint on p. Precisely, we prove that the set U1+epsilon <= p <= q0 E-g(p) is C -compact for any e > 0. The continuity of the map p E [1, qo) -> Bo(p, g) is discussed in detail since it plays a key role in the proof of the main theorem. (C) 2010 Elsevier Inc. All rights reserved.