Vol. 316, No. 2, 2022

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

Hussein Cheikh Ali

Vol. 316 (2022), No. 2, 249–276

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.

Hardy–Sobolev inequality, second best constant, blow-up, optimal inequalities, compact Riemannian manifolds
Mathematical Subject Classification
Primary: 58J05
Received: 17 September 2020
Revised: 27 September 2021
Accepted: 4 December 2021
Published: 6 April 2022
Hussein Cheikh Ali
Département de Mathématique
Université Libre de Bruxelles