Let (M, g) be a compact Riemannian manifold of dimension n ≥ 2 and 1 < p ≤ 2. In this work we prove the validity of the optimal Gagliardo-Nirenberg inequalityfor a family of parameters r, q and θ. Our proof relies strongly on a new distance lemma which holds for 1 < p ≤ 2.In particular, we obtain Riemannian versions of L p -Euclidean Gagliardo-Nirenberg inequalities of [8] and extend the optimal L 2 -Riemannian Gagliardo-Nirenberg inequality of [5] in a unified framework.

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Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F α defined on T M, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g α and of coefficients a α on M, we discuss the compactness problem of minimal energy type solutions of the equationThis question is directly connected to the study of the first best constant A α opt associated with the Riemannian F α -Sobolev inequalityPrecisely, we need to know the dependence of A α opt under F α and g α . For that, we obtain its value as the supremum on M of best constants associated with certain homogeneous Sobolev inequalities on each tangent space and show that A α opt is attained on M. We then establish the continuous dependence of A α opt in relation to F α and g α . The tools used here are based on convex analysis, blow-up, and variational approach.

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