The second best constant for the Hardy–Sobolev inequality on manifolds

Hussein Cheikh Ali

Vol. 316 (2022), No. 2, 249–276

DOI: 10.2140/pjm.2022.316.249

Abstract

We consider the second best constant in the Hardy–Sobolev inequality on a Riemannian manifold. More precisely, we are interested in the existence of extremal functions for this inequality. This problem was tackled by Djadli and Druet (Calc. Var. Partial Differential Equations 12 (2001), 59–84) for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions to Hardy–Sobolev equations of minimizing type. This yields information on the value of the second best constant in the related Riemannian functional inequality.

Keywords

Hardy–Sobolev inequality, second best constant, blow-up, optimal inequalities, compact Riemannian manifolds

Mathematical Subject Classification

Primary: 58J05

Milestones

Received: 17 September 2020

Revised: 27 September 2021

Accepted: 4 December 2021

Published: 6 April 2022