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Non-topological solutions in a generalized Chern-Simons model on torus

We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we show that under doubly periodic condition, there are only two types of solutions, topological and non-topological solutions as the coupling parameter goes to zero. Moreover, we succeed to construct non-topological solution with $k$ bubbles where $k\in\mathbb{N}$ is any given number. We believe that it is the first result for the existence of non-topological doubly periodic solution of the quasi-linear elliptic equation arising in a generalized self-dual Chern-Simons-Higgs gauge theory. To find a suitable approximate solution, it is important to understand the structure of quasi-linear elliptic equation.