Classification and sharp range of flux-pairs for radial solutions to a coupled system

In this paper, we study a coupled system of two nonlinear partial differential equations in the plane, which is related to Liouville equations, non-abelian Higgs BPS vortex equations or two Higgs electroweak model, with singularities at the origin. In addition to deriving the uniqueness of the so-called topological solutions, we also clarify the structure of all types of solutions, including the blow-up ones, under various conditions on coefficients and parameters appearing in the system. Furthermore, the sharp range of flux-pairs associated with specific types of solutions is considered as well.