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Abstract
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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.
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Keywords
Hardy–Sobolev inequality, second best constant, blow-up,
optimal inequalities, compact Riemannian manifolds
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Mathematical Subject Classification
Primary: 58J05
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Milestones
Received: 17 September 2020
Revised: 27 September 2021
Accepted: 4 December 2021
Published: 6 April 2022
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