Ground-state blowing-up solutions for a Hardy-Sobolev equations on a manifold

TL;DR

This paper establishes the existence of minimal blow-up solutions for Hardy-Sobolev equations on manifolds in high dimensions, under a sharp potential condition at the singular point.

Contribution

It demonstrates the existence of blowing-up solutions with minimal energy in Hardy-Sobolev equations on manifolds, extending previous results with a sharp potential condition.

Findings

Existence of blow-up solutions in high dimensions

Solutions are of minimal type

Potential condition is sharp

Abstract

We prove the existence of blowing-up families of solutions to an equation of Hardy-Sobolev type in high dimensions. These families are of minimal type. The sole condition is that the potential of the linear operator touches a critical potential at the singular point. This condition is sharp as shown by the first author in [Cheikh-Ali, Pacific J. of Math. 2022].