This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_\beta\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_\beta}}{|x|^{\beta}} \mathrm{d}x\right)^\frac{p}{p^*_\beta},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_\beta>0$, where $1Comment: 49 pages. arXiv admin note: text overlap with arXiv:2212.05459; text overlap with arXiv:2003.04037 by other authors
