Periodic Maxwell–Chern–Simons vortices with concentrating property

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell–Chern–Simons (MCS) model was introduced by Lee et al. (Phys Lett B 252:79–83, 1990) as a unified system of the classical Abelian–Higgs model (AH) and the Chern–Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern–Simons parameter, without any restriction on either a particular class of solutions or the number of vortex points, as the Chern–Simons mass scale tends to infinity. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by Ricciardi and Tarantello (Comm Pure Appl Math 53:811–851, 2000) and Tarantello (Milan J Math 72:29–80, 2004), and we succeed to establish the existence of periodic Maxwell–Chern–Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.