# Uniqueness of stationary states for singular Keller-Segel type models

## TL;DR

This paper proves the uniqueness of stationary states for a generalized Keller-Segel model with singular potentials, using a new Hardy-Littlewood-Sobolev inequality, in both diffusion-dominated and balanced regimes.

## Contribution

It establishes the first rigorous proof of stationary state uniqueness for singular Keller-Segel models with non-local interactions, extending previous results to more singular potentials.

## Key findings

- Uniqueness of stationary states in singular Keller-Segel models.
- Development of a sharp radial Hardy-Littlewood-Sobolev inequality.
- Applicability in both diffusion-dominated and fair-competition regimes.

## Abstract

We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy-Littlewood-Sobolev type functional inequality in the radial setting.