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

TL;DR

This paper investigates the existence of extremal functions for the second best constant in the Hardy-Sobolev inequality on Riemannian manifolds, extending previous work to the singular case and analyzing blow-up behavior.

Contribution

It establishes the existence of extremal functions for the Hardy-Sobolev inequality's second best constant on manifolds in the singular case and performs a blow-up analysis of solutions.

Findings

Existence of extremal functions for the second best constant.

Blow-up analysis provides insights into the value of the constant.

Extension of results to the singular Hardy-Sobolev case.

Abstract

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.