  This paper is concerned with a study of the classical isoperimetric problem
modified by an addition of a non-local repulsive term. We characterize
existence, non-existence and radial symmetry of the minimizers as a function of
mass in the situation where the non-local term is generated by a kernel given
by an inverse power of the distance. We prove that minimizers of this problem
exist for sufficiently small masses and are given by disks with prescribed mass
below a certain threshold, when the interfacial term in the energy is dominant.
At the same time, we prove that minimizers fail to exist for sufficiently large
masses due to the tendency of the low energy configuration to split into
smaller pieces when the non-local term in the energy is dominant. In the latter
regime, we also establish linear scaling of energy with mass suggesting that
for large masses low energy configurations consist of many roughly equal size
pieces far apart. In the case of slowly decaying kernels we give a complete
characterization of the minimizers.
