Geometry of minimizers for the interaction energy with mildly repulsive   potentials
  We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class $C^2$ and mildly repulsive at the origin; moreover, if the minimizer is global, then its support is finite. In addition, for some class of potentials we prove the validity of a uniform upper bound on the cardinal of the support of a global minimizer. Finally, in the one-dimensional case, we give quantitative bounds. 
J. A. Carrillo, A. Figalli, F. S. Patacchini
math.AP