On a class of Kirchhoff type problems with singular exponential nonlinearity

Abstract

We study the following singular Kirchhoff type problem 
 \[\left( P\right) \left\{
 \begin{array} [c]{c}
 -m\left({\displaystyle\int\limits_{\Omega}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u=h\left( u\right)
 \frac{e^{\alpha u^{2}}}{\left\vert x\right\vert ^{\beta}}\text{ \ \ \ in} \Omega,\\
 u=0 \text{on}\; \partial\Omega
 \end{array} \right.
 \]
 where $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary and $0\in\Omega,$ $\beta\in\left[ 0,2\right)$, $\alpha>0$ and $m$ is a continuous function
 on $\mathbb{R}^{+}.$ Here, $h$ is a suitable preturbation of $e^{\alpha u^{2}}$ as $u\rightarrow\infty.$ In this paper, we prove the existence of solutions of
 $(P)$. Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem