  We consider a class of nonlocal shape optimization problems for sets of fixed
mass where the energy functional is given by an attractive/repulsive
interaction potential in power-law form. We find that the existence of
minimizers of this shape optimization problem depends crucially on the value of
the mass. Our results include existence theorems for large mass and
nonexistence theorems for small mass in the class where the attractive part of
the potential is quadratic. In particular, for the case where the repulsion is
given by the Newtonian potential, we prove that there is a critical value for
the mass, above which balls are the unique minimizers, and below which
minimizers fail to exist. The proofs rely on a relaxation of the variational
problem to bounded densities, and recent progress on nonlocal obstacle
problems.
