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This paper is devoted to considering the following Hardy-Sobolev inequality\[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq\mathcal{S}_\beta\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_\beta}}{|x|^{\beta}}\mathrm{d}x\right)^\frac{p}{p^*_\beta},\quad \forall u\inC^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_\beta>0$, where$1<p<N$, $0\leq \beta<p$, $p^*_\beta=\frac{p(N-\beta)}{N-p}$. Firstly, sincethis problem involves quasilinear operator, we need to establish a compactembedding theorem for some suitable weighted spaces. Moreover, due to the Hardyterm $|x|^{-\beta}$, some new estimates are established. Based on those works,we give the classification to the linearized problem related to the extremalswhich has its own interest such as in blow-up analysis. Then we investigate thegradient stability of above inequality by using spectral estimate combined witha compactness argument, which extends the work of Figalli and Zhang (Duke Math.J., 2022) to a weighted case.

Stability of Hardy-Sobolev inequality involving p-Laplace