Stability of Hardy-Sobolev inequality involving p-Laplace

Abstract

This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_β}}{|x|^β} \mathrm{d}x\right)^\frac{p}{p^*_β},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_β>0$, where $1<p<N$, $0\leq β<p$, $p^*_β=\frac{p(N-β)}{N-p}$. 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 $|x|^{-β}$, 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.