Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetrie decreasing re arrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing ϕ{symbol}. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.