On a System of Nonlinear Elliptic Equations Arising in Theoretical Physics

In this paper we study a system of nonlinear elliptic equations, known as the “vortex equations” in 2 dimensions, arising from the field-theoretical descriptions of several models in physics. When the underlying space is a closed surface, we prove the existence and uniqueness of a solution under a necessary and sufficient condition. When the space is R2, we establish the existence, uniqueness and sharp decay estimates for a solution.