研究期間：10108~10207;The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by [Lin, Ponce and Yang, JFA, 2008]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. For the case of one vortex at origin, in [Chern-Chen-Lin, Comm. Math. Physics, 2010], we proved the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions. We can also classify the structure of all radial solutions. In this project we will investigate the Chern-Simons-Higgs Models. We consider some important cases of the respective elliptic equations, and discuss the properties of the solutions respectively. We will try to investigate the existence and uniqueness of topological solutions, the existence of non-topological solutions. We also want to prove the sharp range of flux and the structure of solutions. Furthermore we intend to establish the uniqueness of solutions with the presence of vortices and anti-vortices.
