The Nehari manifold method for Fractional Kirchhoff problem involving singular and exponential nonlinearity

Abstract: In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left{\begin{split} M\left(|u|^{\frac{n}{s}}\right)(-\Delta)^s_{n/s}u & = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\;\text{in}\;\Om, u&>0,\;\text{in}\; \Om, u &= 0,\;\text{in}\; \mb R^n \setminus{\Om}, \end{split} \right. \end{equation*} where $\Om$ is smooth bounded domain in $\mb R^n$, {$n\geq 1$}, $s\in (0,1)$, $\mu>0$ is a real parameter, $\beta <\frac{n}{n-s}$ and $q\in (0,1)$. We have considered the degenerate Kirchhoff case here and used the Nehari manifold techniques to obtain the results.