Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth

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.