Multiplicity of Solutions for a singular Problem Involving the n-Laplacian

In this paper, we study the following n-Laplacian equation with singular and exponential nonlinearities \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _n u=\lambda u^{-q}+u^{p-1}\frac{e^{u^\beta }}{|x|^\alpha }\quad &{} \text{ in } \Omega ,\\ u>0\quad &{} \text{ in } \Omega ,\\ u=0\quad &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-{\mathrm{\Delta }}_{n}u=\lambda u^{-q}+u^{p-1}\frac{e^{u^{\beta }}}{{|x|}^{\alpha }}\hfill & \text{in}\mathrm{\Omega },\hfill \\ u>0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded domain in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ with smooth boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ , \(n\ge 2\) $n\ge 2$ , \(0<q<1\) $0<q<1$ , \(p>2n\) $p>2n$ , \(\beta \in \left( 1,\frac{n}{n-1}\right) \) $\beta \in \left(1,,,\frac{n}{n-1}\right)$ , \(0<\alpha <n\) $0<\alpha <n$ and \(\lambda >0\) $\lambda >0$ is a parameter. By analyzing the energy functional over the suitable subsets of Nehari manifold, two distinct solutions are obtained.