Uniqueness of positive solutions for singular problems involving the $p$-Laplacian

We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem: $-\Delta_p{u}-\lambda\eta(x)\frac{{u}^{p-1}}{\vert x\vert^p} = \mu\frac{{u}^{p-1}}{\vert x\vert^p}$ on a bounded smooth domain $\Omega\subset\mathbb{R} ^N$ with zero boundary condition. We also characterize all positive solutions of $-\Delta_p{u}=\vert\frac{N-p}{p}\vert^p \frac{u^{p-1}}{\vert x\vert^p}$ in $\mathbb{R} ^N\setminus\{0\}$.