Multiple vortices for a self-dual $Crm P(1)$ Maxwell-Chern-Simons model

In this paper we study the multiple existence of multivortices for the self-dual CP(1) Maxwell-Chern-Simons model. The multivortices for this model correspond to solutions for an elliptic system of two unknown functions with exponential nonlinearity and Dirac measures of positive and negative signs as source terms, defined on a two-dimensional compact manifold such as a flat torus. The existence of at least one multivortex solution for this model was established by D. Chae and H.-S. Nam [Ann. Henri Poincar´e, (2001)]; and in the case that all vortex points have a negative sign, the existence of the second multivortex solution was proved by the second author [Differential Integral Equations, (2004)]. This paper settles the question of multiple existence affirmatively in a more general case, that is, the case of mixed (but different numbers of positive and negative) vortices. It is proved first that the elliptic system arising from this model is equivalent to a fourth-order scalar elliptic equation with exponential nonlinearity. Then it is shown that this fourth-order equation has a nice variational structure enjoying the compactness property, and the desired solutions are obtained as local minima and critical points, via a mountain-pass theorem, of the associated energy functional. Construction of the local minimum solution is a key in the proof.