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An improved Hardy-Sobolev inequality and its application


Authors: Adimurthi, Nirmalendu Chaudhuri and Mythily Ramaswamy
Journal: Proc. Amer. Math. Soc. 130 (2002), 489-505
MSC (1991): Primary 35J30
DOI: https://doi.org/10.1090/S0002-9939-01-06132-9
Published electronically: June 11, 2001
MathSciNet review: 1862130
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Abstract:

For $\Omega \subset \mathbb{R}^{n} , n \geq 2$, 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 $(\frac{1}{\log (1/\vert x\vert)})^{2}$. 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 $1< p < n$.


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Additional Information

Adimurthi
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Bangalore centre, IISc Campus, Bangalore-560012, India
Email: aditi@math.tifrbng.res.in

Nirmalendu Chaudhuri
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore-560012, India
Email: cnirmal@math.iisc.ernet.in

Mythily Ramaswamy
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Bangalore centre, IISc Campus, Bangalore-560012, India
Email: mythily@math.tifrbng.res.in

DOI: https://doi.org/10.1090/S0002-9939-01-06132-9
Keywords: Hardy-Sobolev inequality, eigenvalue, p-laplacian
Received by editor(s): July 5, 2000
Published electronically: June 11, 2001
Additional Notes: The second author was supported in part by CSIR, India.
The third author acknowledges funding from the Indo-French Center for Promotion of Advanced Research, under project 1901-02
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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