  We first show the existence of unique global minimizer of the free energy for
all masses associated to a nonlinear diffusion version of the classical
Keller-Segel model when the diffusion dominates over the attractive force of
the chemoattractant. The strategy uses an approximation of the variational
problem in the whole space by the minimization problem posed on bounded balls
with large radii. We show that all stationary states in a wide class coincide
up to translations with the unique compactly supported radially decreasing and
smooth inside its support global minimizer of the free energy. Our results
complement and show alternative proofs with respect to
\cite{Strohmer,LY,Kim-Yao}.
