Hardy-Sobolev inequalities with remainder terms

Abstract

We prove two Hardy-Sobolev type inequalities in ${\mathcal D}^{1,2}({\mathbb R}^N)$, resp. in $H^1_0(\Omega)$, where $\Omega$ is a bounded domain in ${\mathbb R}^N$, $N\geq 3$. The framework involves the singular potential $\vert x\vert ^{-a}$, with $a\in (0,1)$. Our paper extends previous results established by Bianchi and Egnell [A Note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18–24], resp. by Brezis and Lieb [Inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73–86], corresponding to the case $a=0$.