  This paper deals with the existence and multiplicity of solutions for a class
of Kirchhoff type elliptic system involving the Trudinger-Moser exponential
growth nonlinearities. We first study the existence of solutions for the
following system \begin{eqnarray*}
  \left\{ \arraycolsep=1.5pt
  \begin{array}{ll}
  -\big(a_1+b_1\|u\|^{2(\theta_1-1)}\big)\Delta u= \lambda H_u(x,u,v)\ \ \ &\
\mbox{in}\ \ \ \Omega,\\[2mm]
  -\big(a_2+b_2\|v\|^{2(\theta_2-1)}\big)\Delta v= \lambda H_v(x,u,v)\ \ \ &\
\mbox{in}\ \ \ \Omega,\\[2mm]
  u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partial\Omega,
  \end{array}
  \right.
  \end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with
smooth boundary,\ $\|u\|=\big(\int_{\Omega}|\nabla u|^2dx\big)^{1/2}$, $H_u$
and $H_v$ behave like $e^{\beta |s|^2}$ when $|s|\rightarrow \infty$ for some
$\beta>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $\theta_1,\ \theta_2> 1$ and
$\lambda$ is a positive parameter. In the later part of the paper, we also
discuss a new multiplicity result for the above system with a positive
parameter induced by the nonlocal dependence. The Kirchhoff term and the lack
of compactness of the associated energy functional due to the Trudinger-Moser
embedding have to be overcome via some new techniques.
