  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.
