Stability of Hardy-Sobolev inequality involving p-Laplace

TL;DR

This paper investigates the stability of a Hardy-Sobolev inequality involving the p-Laplace operator, establishing new estimates, compact embeddings, and analyzing the linearized problem to extend stability results to weighted cases.

Contribution

It introduces new estimates and compact embedding theorems for weighted spaces, classifies the linearized problem's extremals, and extends gradient stability analysis to weighted Hardy-Sobolev inequalities.

Findings

Established compact embedding theorems for weighted spaces.

Classified extremals of the linearized problem.

Extended stability results to weighted Hardy-Sobolev inequalities.

Abstract

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