A Kirchhoff type elliptic systems with exponential growth nonlinearities

Abstract

In this paper we are interested in the existence of solutions for the following Kirchhoff type elliptic systems \begin{equation*}\begin{cases}\displaystyle -M\Bigg(\sum^m_{j=1}\|u_j\|^2\Bigg)\Delta u_i=f_i(x,u_1,\ldots,u_m) &\mbox{in } \Omega,\\ u_1=\ldots=u_m=0 &\mbox{on } \partial\Omega, \end{cases}\end{equation*}where $\Omega$ is a bounded domain in $\mathbb{R}^2$, $M$ is a Kirchhoff type function, $\|u_i\|^2:=\int_\Omega|\nabla u_i|^2{d}x$, $f_i$ behaves like $\exp(\beta s^2)$ when $|s|\rightarrow \infty$ for some $\beta> 0$, $i=1,\ldots,m$. By variational methods with the Trudinger-Moser inequality, we obtain the existence of solutions for the above systems.