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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

(1) |

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

**1.**Luis A. Caffarelli, Basilis Gidas, and Joel Spruck,*Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math.**42**(1989), no. 3, 271–297. MR**982351**, 10.1002/cpa.3160420304**2.**Wen Xiong Chen and Congming Li,*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J.**63**(1991), no. 3, 615–622. MR**1121147**, 10.1215/S0012-7094-91-06325-8**3.**Wenxiong Chen and Congming Li,*Regularity of solutions for a system of integral equations*, Commun. Pure Appl. Anal.**4**(2005), no. 1, 1–8. MR**2126275****4.**Wenxiong Chen, Congming Li, and Biao Ou,*Classification of solutions for an integral equation*, Comm. Pure Appl. Math.**59**(2006), no. 3, 330–343. MR**2200258**, 10.1002/cpa.20116**5.**Wenxiong Chen, Congming Li, and Biao Ou,*Classification of solutions for a system of integral equations*, Comm. Partial Differential Equations**30**(2005), no. 1-3, 59–65. MR**2131045**, 10.1081/PDE-200044445**6.**Wenxiong Chen, Congming Li, and Biao Ou,*Qualitative properties of solutions for an integral equation*, Discrete Contin. Dyn. Syst.**12**(2005), no. 2, 347–354. MR**2122171****7.**B. Gidas, Wei Ming Ni, and L. Nirenberg,*Symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ*, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR**634248****8.**Chao Jin and Congming Li,*Symmetry of solutions to some systems of integral equations*, Proc. Amer. Math. Soc.**134**(2006), no. 6, 1661–1670 (electronic). MR**2204277**, 10.1090/S0002-9939-05-08411-X**9.**Chao Jin and Congming Li,*Quantitative analysis of some system of integral equations*, Calc. Var. Partial Differential Equations**26**(2006), no. 4, 447–457. MR**2235882**, 10.1007/s00526-006-0013-5**10.**Congming Li,*Local asymptotic symmetry of singular solutions to nonlinear elliptic equations*, Invent. Math.**123**(1996), no. 2, 221–231. MR**1374197**, 10.1007/s002220050023**11.**Elliott H. Lieb,*Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities*, Ann. of Math. (2)**118**(1983), no. 2, 349–374. MR**717827**, 10.2307/2007032**12.**Congming Li and Jisun Lim,*The singularity analysis of solutions to some integral equations*, Commun. Pure Appl. Anal.**6**(2007), no. 2, 453–464. MR**2289831**, 10.3934/cpaa.2007.6.453**13.**E. M. Stein and Guido Weiss,*Fractional integrals on 𝑛-dimensional Euclidean space*, J. Math. Mech.**7**(1958), 503–514. MR**0098285****14.**Juncheng Wei and Xingwang Xu,*Classification of solutions of higher order conformally invariant equations*, Math. Ann.**313**(1999), no. 2, 207–228. MR**1679783**, 10.1007/s002080050258

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:
http://dx.doi.org/10.1090/S0002-9939-07-09232-5

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.