  We consider macroscopic descriptions of particles where repulsion is modelled
by non-linear power-law diffusion and attraction by a homogeneous singular
kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse
the regime in which diffusive forces are stronger than attraction between
particles, known as the diffusion-dominated regime, and show that all
stationary states of the system are radially symmetric decreasing and compactly
supported. The model can be formulated as a gradient flow of a free energy
functional for which the overall convexity properties are not known. We show
that global minimisers of the free energy always exist. Further, they are
radially symmetric, compactly supported, uniformly bounded and $C^\infty$
inside their support. Global minimisers enjoy certain regularity properties if
the diffusion is not too slow, and in this case, provide stationary states of
the system. In one dimension, stationary states are characterised as optimisers
of a functional inequality which establishes equivalence between global
minimisers and stationary states, and allows to deduce uniqueness.
