On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

Abstract

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -m\left(\sum^k_{j=1}\|u_j\|^2\right)Δu_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^β}+\varepsilon h_i(x),\ \ & \mbox{in}\ \ Ω, \ \ i=1,\ldots,k ,\\[2mm] u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^2$ containing the origin with smooth boundary, $\beta \in [0,2)$, $m$ is a Kirchhoff type function, $\Vert u_j{\Vert }^2={\int }_{\Omega }|\nabla u_j{|}^{2}dx$, $f_i$ behaves like $e^{\beta s^2}$ when $|s|\to \infty$ for some $\beta >0$, and there is $C^1$ function $F:\Omega \times {\mathbb{R}}^k\to \mathbb{R}$ such that $\left(\frac{\partial F}{\partial u_1},\dots ,\frac{\partial F}{\partial u_k}\right)=\left(f_1,\dots ,f_k\right)$, $h_i\in \left((H_0^1(\Omega ){)}^{\ast },\Vert \cdot {\Vert }_{\ast }\right)$. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when $\epsilon >0$ is small.