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: {Delta u + 1/epsilon(2) e(u)(1 - e(u)) = 4 pi Sigma(N)(j=1) delta(pj), in Omega, u is doubly periodic on partial derivative Omega, (0.1) where Omega is a parallelogram in R-2 and epsilon > 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). (C) 2018 Elsevier Inc. All rights reserved.