Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Wenxiong Chen; Congming Li
Journal: Proc. Amer. Math. Soc. 136 (2008), 955-962.
MSC (2000): Primary 35J45, 35J60; Secondary 45G05, 45G15
Posted: November 30, 2007
MathSciNet review: 2361869
Retrieve article in: PDF

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 , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

References:

1.: L. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. XLII, (1989), 271-297. MR 982351 (90c:35075)
2.: W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. MR 1121147 (93e:35009)
3.: W. Chen and C. Li, Regularity of Solutions for a system of Integral Equations, Comm. Pure and Appl. Anal., 4(2005), 1-8. MR 2126275 (2006g:45006)
4.: W. Chen, C. Li, and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59(2006), 330-343. MR 2200258 (2006m:45007a)
5.: W. Chen, C. Li, and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30(2005), 59-65. MR 2131045 (2006a:45007)
6.: W. Chen, C. Li, and B. Ou, Qualitative Properties of Solutions for an Integral Equation, Disc. & Cont. Dynamics Sys., 12(2005), 347-354. MR 2122171 (2006g:45009)
7.: B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^{n},$ Mathematical Analysis and Applications, Academic Press, New York, 1981. MR 634248 (84a:35083)
8.: C. Jin and C. Li, Symmetry of Solutions to Some Integral Equations, Proc. Amer. Math. Soc., 134(2006), 1661-1670. MR 2204277 (2006j:45017)
9.: C. Jin and C. Li, Quantitative Analysis of Some System of Integral Equations, Cal. Var. PDEs, 26(2006), 447-457. MR 2235882 (2007c:45013)
10.: C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. MR 1374197 (96m:35085)
11.: E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118(1983), 349-374. MR 717827 (86i:42010)
12.: C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal. 6 (2007), 453-464. MR 2289831
13.: E. M. Stein and G. Weiss, Fractional integrals in -dimensional Euclidean space J. Math. Mech., 7 (1958). MR 0098285 (20:4746)
14.: J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. MR 1679783 (2000a:58093)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J45, 35J60, 45G05, 45G15

Retrieve articles in all Journals with MSC (2000): 35J45, 35J60, 45G05, 45G15

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: 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
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|