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

Abstract

In this paper, we study the following n-Laplacian equation with singular and exponential nonlinearities

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _n u=\lambda u^{-q}+u^{p-1}\frac{e^{u^\beta }}{|x|^\alpha }\quad &{} \text{ in } \Omega ,\\ u>0\quad &{} \text{ in } \Omega ,\\ u=0\quad &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\) with smooth boundary \(\partial \Omega \), \(n\ge 2\), \(0<q<1\), \(p>2n\), \(\beta \in \left( 1,\frac{n}{n-1}\right) \), \(0<\alpha <n\) and \(\lambda >0\) is a parameter. By analyzing the energy functional over the suitable subsets of Nehari manifold, two distinct solutions are obtained.