# An Improved Hardy-Sobolev Inequality and its Application

Adimurthi, * and Chaudhuri, Nirmalendu and Ramaswamy, Mythily (2001) An Improved Hardy-Sobolev Inequality and its Application. In: Proceedings of the American Mathematical Society, 130 (2). pp. 489-505.

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## Abstract

For\Omega \subset $IR^n$,n\geq 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality, by adding a term with a singular weight of the type \frac{1}{log(1/|x|)}$^2$ . 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\omega := -(div(|\nabla\upsilon|{p-2}\nabla\upilson)as \mu increases to \frac{n-p}{p}$^p$ for 1 < p < n.

Item Type: Journal Article Copyright of this article belongs to American Mathematical Society. Hardy-Sobolev inequality;eigenvalue;p-laplacian Division of Physical & Mathematical Sciences > Mathematics Ravi V Nandhan 24 Jan 2005 01 Mar 2012 09:09 http://eprints.iisc.ac.in/id/eprint/1852