  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.
