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An improved Hardy-Sobolev inequality and its application

Authors:	Adimurthi Nirmalendu Chaudhuri Mythily Ramaswamy

Institution:	School of Mathematics, Tata Institute of Fundamental Research, Bangalore centre, IISc Campus, Bangalore-560012, India ; Department of Mathematics, Indian Institute of Science, Bangalore-560012, India ; School of Mathematics, Tata Institute of Fundamental Research, Bangalore centre, IISc Campus, Bangalore-560012, India

Abstract:	For $\Omega \subset \mathbb{R}^{n} , n \geq 2$ , a bounded domain, and for , we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type $(\frac{1}{\log (1/\vert x\vert)})^{2}$ . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator $L_{\mu }u:= - (\text{div}(\vert\nabla u\vert^{p-2}\nabla u) + \frac{\mu }{\vert x\vert^{p}} \vert u\vert^{p-2}u )$ as $\mu$ increases to $\left (\frac{n-p}{p}\right )^{p}$ for .

Keywords:	Hardy-Sobolev inequality eigenvalue p-laplacian

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