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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The best constant in a weighted Hardy-Littlewood-Sobolev inequality


Authors: Wenxiong Chen and Congming Li
Journal: Proc. Amer. Math. Soc. 136 (2008), 955-962
MSC (2000): Primary 35J45, 35J60; Secondary 45G05, 45G15
Published electronically: November 30, 2007
MathSciNet review: 2361869
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the uniqueness for the solutions of the singular nonlinear PDE system:

$\displaystyle \left\{\begin{array}{ll} - \lap ( \vert x\vert^{\alpha} u(x) ) = ... ...rt^{\beta} v(x) ) = \dfrac{u^p (x)}{\vert x\vert^{\alpha}}. \end{array} \right.$ (1)

In the special case when $ \alpha = \beta$ and $ p = q$, we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.


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Additional Information

Wenxiong Chen
Affiliation: College of Mathematics and Information Science, Henan Normal University, People’s Republic of China
Address at time of publication: Department of Mathematics, Yeshiva University, 500 W. 185th Street, New York, New York 10033
Email: wchen@yu.edu

Congming Li
Affiliation: Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309
Email: cli@colorado.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09232-5
PII: S 0002-9939(07)09232-5
Keywords: Weighted Hardy-Littlewood-Sobolev inequality, best constants, system of singular PDEs, uniqueness, radial symmetry, classifications
Received by editor(s): November 13, 2006
Published electronically: November 30, 2007
Additional Notes: The first author was partially supported by NSF Grant DMS-0604638
The second author was partially supported by NSF Grant DMS-0401174
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.