Existence of multiple positive solutions for a nonlinear elliptic problem with the critical exponent and a Hardy term

Abstract

In this paper, we show that if $\mu>0$ is small enough, the problem \begin{equation*} \left\{ \begin{aligned} -\Delta u -\mu\frac{u}{|x|^2} & =|u|^{2^\ast-2}u & & \text{in $\Omega$,}\\ u & =0 & & \text{on $\partial\Omega$} \end{aligned} \right. \end{equation*} has at least cat $\Omega -1$ positive solutions, where $\Omega$ is a noncontractible, bounded domain in $\mathbb R^N (N\geq 4)$ such that its boundary is smooth and $0 \in \Omega$.