Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications

Abstract

Let ( M, g) be a smooth compact Riemannian manifold, with or without boundary, of dimension n⩾3 and 1< p< n/2. Considering the norm ‖u‖= ‖ Δ gu‖ L p(M) p+‖u‖ L p(M) p 1/p on each of the spaces H 2, p ( M), H 0 2, p ( M) and H 2, p ( M)∩ H 0 1, p ( M), we study an asymptotically sharp inequality associated to the critical Sobolev embedding of these spaces. As an application, we investigate the influence of the geometry in the existence of solutions for some fourth-order problems involving critical exponents on manifolds. In particular, new phenomena arise in Brezis–Nirenberg type problems on manifolds with positive scalar curvature somewhere, in contrast with the Euclidean case. We also show that on such manifolds the corresponding optimal inequality for p=2 is not valid.