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.