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 , Djadli and Druet (2001) [13] proved the existence of extremal functions to the following sharp Riemannian Lp-Sobolev inequality: where and K(n,p)p and B0(p,g) stands for, respectively, the first and second Sobolev best constants for this inequality. Let then Eg(p) be the corresponding extremal set normalized by the unity Lp*-norm. In contrast what happens in the whole space for 10. The continuity of the map p [1,q0) B0(p,g) is discussed in detail since it plays a key role in the proof of the main theorem.