Symmetry of solutions to some systems of integral equations
Authors:
Chao Jin and Congming Li
Journal:
Proc. Amer. Math. Soc. 134 (2006), 16611670
MSC (2000):
Primary 35J99, 45E10, 45G05
Published electronically:
October 28, 2005
MathSciNet review:
2204277
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In this paper, we study some systems of integral equations, including those related to HardyLittlewoodSobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the EulerLagrange equations associated with the classical and weighted HardyLittlewoodSobolev inequality.
 1.
H.
Berestycki and L.
Nirenberg, On the method of moving planes and the sliding
method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991),
no. 1, 1–37. MR 1159383
(93a:35048), http://dx.doi.org/10.1007/BF01244896
 2.
Haïm
Brézis and Tosio
Kato, Remarks on the Schrödinger operator with singular
complex potentials, J. Math. Pures Appl. (9) 58
(1979), no. 2, 137–151. MR 539217
(80i:35135)
 3.
Haïm
Brezis and Elliott
H. Lieb, Minimum action solutions of some vector field
equations, Comm. Math. Phys. 96 (1984), no. 1,
97–113. MR
765961 (86d:35045)
 4.
William
Beckner, Sharp Sobolev inequalities on the sphere and the
MoserTrudinger inequality, Ann. of Math. (2) 138
(1993), no. 1, 213–242. MR 1230930
(94m:58232), http://dx.doi.org/10.2307/2946638
 5.
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
(90c:35075), http://dx.doi.org/10.1002/cpa.3160420304
 6.
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
(93e:35009), http://dx.doi.org/10.1215/S0012709491063258
 7.
Wenxiong
Chen and Congming
Li, A priori estimates for prescribing scalar curvature
equations, Ann. of Math. (2) 145 (1997), no. 3,
547–564. MR 1454703
(98d:53049), http://dx.doi.org/10.2307/2951844
 8.
W. Chen, C. Li, and B. Ou, Classification of solutions for an integral equation, to appear Comm. Pure and Appl. Math.
 9.
Wenxiong
Chen, Congming
Li, and Biao
Ou, Classification of solutions for a system of integral
equations, Comm. Partial Differential Equations 30
(2005), no. 13, 59–65. MR 2131045
(2006a:45007), http://dx.doi.org/10.1081/PDE200044445
 10.
A. Chang and P. Yang, On uniqueness of an nth order differential equation in conformal geometry, Math. Res. Letters, 4(1997), 112.
 11.
L.
E. Fraenkel, An introduction to maximum principles and symmetry in
elliptic problems, Cambridge Tracts in Mathematics, vol. 128,
Cambridge University Press, Cambridge, 2000. MR 1751289
(2001c:35042)
 12.
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, 1981, pp. 369–402. MR 634248
(84a:35083)
 13.
Congming
Li, Local asymptotic symmetry of singular solutions to nonlinear
elliptic equations, Invent. Math. 123 (1996),
no. 2, 221–231. MR 1374197
(96m:35085), http://dx.doi.org/10.1007/s002220050023
 14.
Elliott
H. Lieb, Sharp constants in the HardyLittlewoodSobolev and
related inequalities, Ann. of Math. (2) 118 (1983),
no. 2, 349–374. MR 717827
(86i:42010), http://dx.doi.org/10.2307/2007032
 15.
Elliott
H. Lieb and Michael
Loss, Analysis, 2nd ed., Graduate Studies in Mathematics,
vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225
(2001i:00001)
 16.
Biao
Ou, A remark on a singular integral equation, Houston J. Math.
25 (1999), no. 1, 181–184. MR 1675383
(2000e:45004)
 17.
James
Serrin, A symmetry problem in potential theory, Arch. Rational
Mech. Anal. 43 (1971), 304–318. MR 0333220
(48 #11545)
 18.
Elias
M. Stein, Singular integrals and differentiability properties of
functions, Princeton Mathematical Series, No. 30, Princeton University
Press, Princeton, N.J., 1970. MR 0290095
(44 #7280)
 19.
Juncheng
Wei and Xingwang
Xu, Classification of solutions of higher order conformally
invariant equations, Math. Ann. 313 (1999),
no. 2, 207–228. MR 1679783
(2000a:58093), http://dx.doi.org/10.1007/s002080050258
 1.
 H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.) 22 (1) (1991), 137. MR 1159383 (93a:35048)
 2.
 H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pure Appl. 58 (2) (1979), 137151. MR 0539217 (80i:35135)
 3.
 H. Brezis and E. H. Lieb, Minimum action of some vectorfield equations, Commun. Math. Phys. 96 (1) (1984), 97113. MR 0765961 (86d:35045)
 4.
 W. Beckner, Sharp Sobolev inequalities on the sphere and the MoserTrudinger inequality, Ann. of Math. 138(1993) 213242. MR 1230930 (94m:58232)
 5.
 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), 271297. MR 0982351 (90c:35075)
 6.
 W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615622. MR 1121147 (93e:35009)
 7.
 W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145(1997), 547564. MR 1454703 (98d:53049)
 8.
 W. Chen, C. Li, and B. Ou, Classification of solutions for an integral equation, to appear Comm. Pure and Appl. Math.
 9.
 W. Chen, C. Li, and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30(2005) 5965. MR 2131045
 10.
 A. Chang and P. Yang, On uniqueness of an nth order differential equation in conformal geometry, Math. Res. Letters, 4(1997), 112.
 11.
 L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Unversity Press, New York, 2000. MR 1751289 (2001c:35042)
 12.
 B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in (collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981). MR 0634248 (84a:35083)
 13.
 C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math. 123(1996) 221231. MR 1374197 (96m:35085)
 14.
 E. Lieb, Sharp constants in the HardyLittlewoodSobolev and related inequalities, Ann. of Math. 118(1983), 349374. MR 0717827 (86i:42010)
 15.
 E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Rhode Island, 2001. MR 1817225 (2001i:00001)
 16.
 B. Ou, A Remark on a singular integral equation, Houston J. of Math. 25 (1) (1999), 181  184. MR 1675383 (2000e:45004)
 17.
 J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304318 (1971). MR 0333220 (48:11545)
 18.
 E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. MR 0290095 (44:7280)
 19.
 J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313, (1999) 207228 . MR 1679783 (2000a:58093)
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Additional Information
Chao Jin
Affiliation:
Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, Colorado 80309
Email:
jinc@colorado.edu
Congming Li
Affiliation:
Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, Colorado 80309
Email:
cli@colorado.edu
DOI:
http://dx.doi.org/10.1090/S000299390508411X
PII:
S 00029939(05)08411X
Keywords:
HardyLittlewoodSobolev inequalities,
systems of integral equations,
radial symmetry,
classification of solution
Received by editor(s):
July 28, 2004
Received by editor(s) in revised form:
December 29, 2004
Published electronically:
October 28, 2005
Additional Notes:
This work was partially supported by NSF grant DMS0401174.
Communicated by:
David S. Tartakoff
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
