Bivariate Hardy-Sobolev Inequality and Its Sharp Stability

Abstract

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ ($N\ge 3$) be an open domain, $s\in (0,2)$, $\hat{I}\pm >1$, ${\hat{I}}^2>1$ with $\hat{I}\pm +{\hat{I}}^2=2^{\ast }(s)$, and $\hat{I}º\in \mathbb{R}$. For any functions $u,v\in D_0^{1,2}(\hat{I}\text{c◯})$, we prove the inequality: \begin{multline*} \int_Ω |\nabla u|^2 \, \mathrm{d}x + \int_Ω |\nabla v|^2 \, \mathrm{d}x \ge S_{α,β,λ,μ}(Ω) \left( \int_Ω \Big( λ\frac{|u|^{2^*(s)}}{|x|^s} + μ\frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) κ\frac{|u|^α|v|^β}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{\hat{I}\pm ,{\hat{I}}^2,\hat{I}»,\hat{I}¼}(\hat{I}\text{c◯})$ and characterize the set of minimizers. Moreover, for $\hat{I}\text{c◯}={\mathbb{R}}^N$ and $\hat{I}º>0$, we obtain sharp stability results for nonnegative functions.