Bivariate Hardy-Sobolev Inequality and Its Sharp Stability

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(\Omega)$, we prove the inequality: \begin{multline*} \int_{\Omega} |\nabla u|^2 \, \mathrm{d}x + \int_{\Omega} |\nabla v|^2 \, \mathrm{d}x \ge S_{\alpha,\beta,\lambda,\mu}(\Omega) \left( \int_{\Omega} \Big( \lambda \frac{|u|^{2^*(s)}}{|x|^s} + \mu \frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) \kappa \frac{|u|^\alpha |v|^\beta}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{\alpha,\beta,\lambda,\mu}(\Omega)$ and characterize the set of minimizers. Moreover, for $\Omega = \mathbb{R}^N$ and $\kappa > 0$, we obtain sharp stability results for nonnegative functions.