  We consider a nonlocal aggregation equation with degenerate diffusion, which
describes the mean-field limit of interacting particles driven by nonlocal
interactions and localized repulsion. When the interaction potential is
attractive, it is previously known that all steady states must be radially
decreasing up to a translation, but uniqueness (for a given mass) within the
radial class was open, except for some special interaction potentials. For
general attractive potentials, we show that the uniqueness/non-uniqueness
criteria are determined by the power of the degenerate diffusion, with the
critical power being $m = 2$. In the case $m \ge 2$, we show that for any
attractive potential the steady state is unique for a fixed mass. In the case
$1 < m < 2$, we construct examples of smooth attractive potentials, such that
there are infinitely many radially decreasing steady states of the same mass.
For the uniqueness proof, we develop a novel interpolation curve between two
radially decreasing densities, and the key step is to show that the interaction
energy is convex along this curve for any attractive interaction potential,
which is of independent interest.
