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.