  We show that partial mass concentration can happen for stationary solutions
of aggregation-diffusion equations with homogeneous attractive kernels in the
fast diffusion range. More precisely, we prove that the free energy admits a
radial global minimizer in the set of probability measures which may have part
of its mass concentrated in a Dirac delta at a given point. In the case of the
quartic interaction potential, we find the exact range of the diffusion
exponent where concentration occurs in space dimensions $N\geq6$. We then
provide numerical computations which suggest the occurrence of mass
concentration in all dimensions $N\geq3$, for homogeneous interaction
potentials with higher power.
