### Hardy-Sobolev inequalities with remainder terms

DOI: http://dx.doi.org/10.12775/TMNA.2002.030

#### 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 [< i> A Note on the Sobolev inequality< /i> , J. Funct. Anal. < b> 100< /b>

(1991), 18–24],

resp. by Brezis and Lieb

[< i> Inequalities with remainder terms< /i> , J. Funct. Anal. < b> 62< /b> (1985),

73–86], corresponding to the case $a=0$.

$H^1_0(\Omega)$, 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 [< i> A Note on the Sobolev inequality< /i> , J. Funct. Anal. < b> 100< /b>

(1991), 18–24],

resp. by Brezis and Lieb

[< i> Inequalities with remainder terms< /i> , J. Funct. Anal. < b> 62< /b> (1985),

73–86], corresponding to the case $a=0$.

#### Keywords

Hardy-Sobolev inequality; minimization problem; singular potential; Schwartz symmetrization

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