# Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds

## TL;DR

This paper investigates the existence of extremal functions for Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds using the Concentration-Compactness principle.

## Contribution

It extends the analysis of Hardy-Littlewood-Sobolev inequalities to the setting of compact Riemannian manifolds and establishes existence results for extremal functions.

## Key findings

- Existence of extremal functions on compact manifolds.
- Application of Concentration-Compactness principle to this problem.
- Extension of classical inequalities to geometric settings.

## Abstract

This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.