  We study the existence and uniqueness of nontrivial stationary solutions to a
nonlocal aggregation equation with quadratic diffusion arising in many contexts
in population dynamics. The equation is the Wasserstein gradient flow generated
by the energy E, which is the sum of a quadratic free energy and the
interaction energy. The interaction kernel is taken radial and attractive,
nonnegative and integrable, with further technical smoothness assumptions. The
existence vs. nonexistence of such solutions is ruled by a threshold
phenomenon, namely nontrivial steady states exist if and only if the
diffusivity constant is strictly smaller than the total mass of the interaction
kernel. In the one dimensional case we prove that steady states are unique up
to translations and mass constraint. The strategy is based on a strong version
of the Krein-Rutman theorem. The steady states are symmetric with respect to
their center of mass x0, compactly supported on sets of the form [x0 -L, x0+L],
C^2 on their support, strictly decreasing on (x0, x0+L). Moreover, they are
global minimizers of the energy functional E. The results are complemented by
numerical simulations.
