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.

We show that the Lp Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.2010 Mathematics Subject Classification. Primary 46E35, Secondary 51M16.

No abstract

In the present work, we consider elliptic systems involving polyharmonic operators and critical exponents. We discuss the existence and nonexistence of nontrivial solutions to these systems. Our theorems improve and/or extend the ones established by Bartsch and Guo [T. Bartsch, Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differential Equations 220 (2006) 531-543] in both aspects of spectral interaction and regularity of lower order perturbations.

We concerns here with the continuity on the geometry of the second Riemannian L p -Sobolev best constant B 0 (p, g) associated to the AB program. Precisely, for 1 ≤ p ≤ 2, we prove that B 0 (p, g) depends continuously on g in the C 2 -topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C 0 -compactness results on extremal functions. * 2000 Mathematics Subject Classification: 32Q10, 53C21 †