# Blow-up analysis for a Hardy-Sobolev equation on compact Riemannian manifolds with application to the existence of solutions

## TL;DR

This paper analyzes a singular elliptic equation with critical Sobolev and Hardy potentials on compact Riemannian manifolds, providing a decomposition of Palais-Smale sequences and establishing existence of solutions at various energy levels.

## Contribution

It introduces a new $H^2_1$ decomposition for Palais-Smale sequences and applies it to prove the existence of solutions with different energies.

## Key findings

- Decomposition of Palais-Smale sequences on manifolds
- Existence of solutions at multiple energy levels
- Application to Hardy-Sobolev equations

## Abstract

On a compact Riemannian manifold, we study a singular elliptic equation with critical Sobolev exponent and critical Hardy potential. In a first part, we prove an $H^2_1$ type decomposition result for Palais-Smale sequences of the associated energy functional. In a second part, we apply the decomposition result to obtain solutions of different energy levels.