On the existence and boundedness of minimizing measures for a general form of non-local energies

In this paper we consider a very general form of a non-local energy in integral form, which covers most of the usual ones (for instance, the sum of a positive and a negative power). Instead of admitting only sets, or \(L^\infty \) $L^{\infty }$ functions, as admissible objects, we define the energy for all the Radon measures. We prove the existence of optimal measures in a wide generality, and we show that in several cases the optimal measures are actually \(L^\infty \) $L^{\infty }$ functions, providing an a priori bound on their norm. We also derive a uniqueness result for minimizers.