On the compactness problem of extremal functions to sharp Riemannian [formula omitted]-Sobolev inequalities

Abstract

Let ( M , g ) be a smooth compact Riemannian manifold without boundary of dimension n ⩾ 2 . For 1 < p < q 0 = min { 2 , n } , Djadli and Druet (2001) [13] proved the existence of extremal functions to the following sharp Riemannian L p -Sobolev inequality: ‖ u ‖ L p ∗ ( M ) p ⩽ K ( n , p ) p ‖ ∇ u ‖ L p ( M ) p + B 0 ( p , g ) ‖ u ‖ L p ( M ) p , where p ∗ = n p 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 n 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 < q 0 . Moreover, we address the question from a uniform viewpoint on p. Precisely, we prove that the set ⋃ 1 + ε ⩽ p ⩽ q 0 − ε E g ( p ) is C 0 -compact for any ε > 0 . The continuity of the map p ∈ [ 1 , q 0 ) ↦ B 0 ( p , g ) is discussed in detail since it plays a key role in the proof of the main theorem.