Bivariate Hardy-Sobolev Inequality and Its Sharp Stability

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be an open domain, $s\in (0,2)$, $\alpha >1$, $\beta >1$ with $\alpha +\beta =2^{\ast }(s)$, and $\kappa \in \mathbb{R}$. For any functions $u,v\in D_0^{1,2}(\mathrm{\Omega })$, 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 $S_{\alpha ,\beta ,\lambda ,\mu }(\mathrm{\Omega })$ and characterize the set of minimizers. Moreover, for $\mathrm{\Omega }={\mathbb{R}}^N$ and $\kappa >0$, we obtain sharp stability results for nonnegative functions.