Sharp existence theorems for multiple vortices induced from magnetic impurities

In a recent work Tong and Wong derived self-dual equations governing multiple vortices in a product Abelian Higgs model which may be regarded as a generalized Ginzburg–Landau theory. The purpose of this paper is to establish some sharp existence and uniqueness theorems for these multiple vortex solutions in a doubly periodic domain and on the full plane using the methods of calculus of variations.