On the mean field type bubbling solutions for Chern-Simons-Higgs equation

This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern-Simons-Higgs equation:(0.1){Δu+1ε2eu(1−eu)=4π∑j=1Nδpj,inΩ,u is doubly periodic on∂Ω, where Ω is a parallelogram in R2 and ε>0 is a small parameter. In part 1 [29], we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern-Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27]. In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1).