Existence of the topological solutions arising in vortices–antivortices equation

In this paper, we first introduce the first-order formalism with the self-dual structure of the motion equations, also called the Bogomol’nyi and Prasad–Sommerfield (BPS) equations. We observe that BPS equations arising in the generalized Maxwell–Higgs model under specific circumstances can be transformed into a vortices–antivortices equation, which is a nonlinear elliptic equation with the exponential functions. For the vortices–antivortices equation in two space dimensions, we prove the existence of topological solutions by a monotone iteration method. Finally, we give the asymptotic estimates of solutions obtained at infinity.