Stability of Hardy-Sobolev inequality involving p-Laplace

This paper is devoted to considering the following Hardy-Sobolev inequality $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p\mathrm{d}x\ge {\mathcal{S}}_{\beta }{\left({\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }u{\mathrm{\mid }}^{p_{\beta }^{\ast }}}{\mathrm{\mid }x{\mathrm{\mid }}^{\beta }}\mathrm{d}x\right)}^{\frac{p}{p_{\beta }^{\ast }}},\mathrm{\forall }u\in C_0^{\mathrm{\infty }}({\mathbb{R}}^N),$$ for some constant ${\mathcal{S}}_{\beta }>0$, where $1<p<N$, $0\le \beta <p$, $p_{\beta }^{\ast }=\frac{p(N-\beta )}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }$, some new estimates are established. Based on those works, we give the classification to the linearized problem related to the extremals which has its own interest such as in blow-up analysis. Then we investigate the gradient stability of above inequality by using spectral estimate combined with a compactness argument, which extends the work of Figalli and Zhang (Duke Math. J., 2022) to a weighted case.