On the Chern-simons-higgs Equation: Part Ii, Local Uniqueness and Exact Number of Solutions

This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern-Simons-Higgs equation: { ∆u+ 1 ε2 e (1− e) = 4π ∑N j=1 δpj , in Ω, u is doubly periodic on ∂Ω, (0.1) where Ω is a parallelogram in R and ε > 0 is a small parameter. In part 1 [28], we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result under some nearly necessary conditions. In this paper, we shall study two other important issues: the local uniqueness of bubbling solutions and the exact number of solutions. The local uniqueness asserts that if uε,1 and uε,2 are two sequence of bubbling solutions blowing up at the same points satisfying some non-degenerate condition, then uε,1 = uε,2 for small ε > 0. The local uniqueness is an important problem not only for the Chern-Simons equation, but for all equations possessing the concentration phenomenon. The idea developed in this paper will provide an effective method to deal with all such equations. A consequence of this local uniqueness result is the establishment of an one to one relation between the bubbling solutions of the above nonlinear PDE and the solutions of some algebraic equations, which enables us to count the exact number of solutions for (0.1) by using the elliptic functions theories if all the vortex points pj collapse to one point, Ω is a rectangle, and 1 ≤ N ≤ 5.