An integral equation on half space
Authors:
Dongyan Li and Ran Zhuo
Journal:
Proc. Amer. Math. Soc. 138 (2010), 27792791
MSC (2010):
Primary 35J99, 45E10, 45G05
Published electronically:
April 14, 2010
MathSciNet review:
2644892
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Additional Information
Abstract: Let be the dimensional upper half Euclidean space, and let be any real number satisfying In this paper, we consider the integral equation  (1)  where , and is the reflection of the point about the hyperplane . We use a new type of moving plane method in integral forms introduced by Chen, Li and Ou to establish the regularity and rotational symmetry of the solution of the above integral equation.
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Congming
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equations, Commun. Pure Appl. Anal. 6 (2007),
no. 2, 453–464. MR 2289831
(2008e:45008), 10.3934/cpaa.2007.6.453
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Li
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Chen, A Liouville type theorem for an integral system, Commun.
Pure Appl. Anal. 5 (2006), no. 4, 855–859. MR 2246012
(2007d:35094), 10.3934/cpaa.2006.5.855
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Li
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Chen, Radial symmetry and monotonicity for an integral
equation, J. Math. Anal. Appl. 342 (2008),
no. 2, 943–949. MR 2445251
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Ma and Lin
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coupled nonlinear Schrödinger system, J. Math. Phys.
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(2009g:35308), 10.1063/1.2939238
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Biao
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25 (1999), no. 1, 181–184. MR 1675383
(2000e:45004)
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Serrin, A symmetry problem in potential theory, Arch. Rational
Mech. Anal. 43 (1971), 304–318. MR 0333220
(48 #11545)
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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), 10.1007/s002080050258
 [BN]
 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)
 [CGS]
 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 982351 (90c:35075)
 [CJ]
 C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 16611670. MR 2204277 (2006j:45017)
 [CJ1]
 C. Jin and C. Li, Quantitative analysis of some system of integral equations, Cal. Var. PDEs, 26 (2006), 447457. MR 2235882 (2007c:45013)
 [CL]
 W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615622. MR 1121147 (93e:35009)
 [CL1]
 W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math. (2), 145 (1997), 547564. MR 1454703 (98d:53049)
 [CL2]
 W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 18. MR 2126275 (2006g:45006)
 [CL3]
 W. Chen and C. Li, The best constant in some weighted HardyLittlewoodSobolev inequality. Proc. Amer. Math. Soc., 136 (2008), 955962. MR 2361869 (2009b:35098)
 [CLO]
 W. Chen, C. Li, and Biao Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330343. MR 2200258 (2006m:45007a)
 [CLO1]
 W. Chen, C. Li, and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347354. MR 2122171 (2006g:45009)
 [CLO2]
 W. Chen, C. Li, and B. Ou, Classification of solutions for a system of integral equations, Comm. PDE, 30 (2005), 5965. MR 2131045 (2006a:45007)
 [CY]
 A. Chang and P. Yang, On uniqueness of an nth order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 112.
 [F]
 L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, Cambridge, 2000. MR 1751289 (2001c:35042)
 [GNN]
 B. Gidas, W.M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in , in Mathematical Analysis and Applications, vol. 7a of Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. MR 634248 (84a:35083)
 [L]
 E. Lieb, Sharp constants in the HardyLittlewoodSobolev and related inequalities, Ann. of Math., 118 (1983), 349374. MR 717827 (86i:42010)
 [Li]
 C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221231. MR 1374197 (96m:35085)
 [LiM]
 C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 10491057. MR 2452879 (2009k:35079)
 [LLim]
 C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure and Applied Analysis, 2 (6) (2007), 112. MR 2289831 (2008e:45008)
 [MC]
 L. Ma and D.Z. Chen, A Liouville type theorem for an integral system, Comm. Pure and Applied Analysis, 5 (2006), 855859. MR 2246012 (2007d:35094)
 [MC2]
 L. Ma and D. Z Chen, Radial symmetry and monotonicity for an integral equation, Journal of Mathematical Analysis and Applications, 342 (2008), 943949. MR 2445251 (2009m:35151)
 [MZ]
 L. Ma and L. Zhao, Sharp thresholds of blowup and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys., 49 (2008), no. 6, 062103, 17 pp. MR 2431772 (2009g:35308)
 [O]
 B. Ou, A remark on a singular integral equation, Houston J. of Math., 25 (1) (1999), 181184. MR 1675383 (2000e:45004)
 [Se]
 J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304318. MR 0333220 (48:11545)
 [WX]
 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
Dongyan Li
Affiliation:
College of Mathematics and Information Science, Henan Normal University, Henan, People’s Republic of China
Email:
w408867388w@126.com
Ran Zhuo
Affiliation:
College of Mathematics and Information Science, Henan Normal University, Henan, People’s Republic of China
Email:
zhuoran1986@126.com
DOI:
http://dx.doi.org/10.1090/S0002993910103682
Keywords:
Integral equations,
regularity,
method of moving planes,
rotational symmetry,
upper half space,
monotonicity.
Received by editor(s):
September 25, 2009
Published electronically:
April 14, 2010
Communicated by:
Matthew J. Gursky
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
