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