  We prove reversed Hardy-Littlewood-Sobolev inequalities by carefully studying
the natural associated free energies with direct methods of calculus of
variations. Tightness is obtained by a dyadic argument, which quantifies the
relative strength of the entropy functional versus the interaction energy. The
existence of optimizers is shown in the class of $\prob$. With respect to their
regularity, we study conditions for optimizers to be bounded functions. In a
related model, we show the condensation phenomena, which suggests that
optimizers are not in general regular.
