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
Fulltext PDF
Abstract 
References 
Similar Articles 
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.
 [BN]
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
 [CGS]
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
 [CJ]
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
(2006j:45017), http://dx.doi.org/10.1090/S000299390508411X
 [CJ1]
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
(2007c:45013), http://dx.doi.org/10.1007/s0052600600135
 [CL]
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
 [CL1]
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
 [CL2]
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
(2006g:45006)
 [CL3]
Wenxiong
Chen and Congming
Li, The best constant in a weighted
HardyLittlewoodSobolev inequality, Proc.
Amer. Math. Soc. 136 (2008), no. 3, 955–962. MR 2361869
(2009b:35098), http://dx.doi.org/10.1090/S0002993907092325
 [CLO]
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
(2006m:45007a), http://dx.doi.org/10.1002/cpa.20116
 [CLO1]
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
(2006g:45009)
 [CLO2]
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
 [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.
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)
 [GNN]
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 YorkLondon, 1981, pp. 369–402. MR 634248
(84a:35083)
 [L]
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
 [Li]
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
 [LiM]
Congming
Li and Li
Ma, Uniqueness of positive bound states to Schrödinger systems
with critical exponents, SIAM J. Math. Anal. 40
(2008), no. 3, 1049–1057. MR 2452879
(2009k:35079), http://dx.doi.org/10.1137/080712301
 [LLim]
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
(2008e:45008), http://dx.doi.org/10.3934/cpaa.2007.6.453
 [MC]
Li
Ma and Dezhong
Chen, A Liouville type theorem for an integral system, Commun.
Pure Appl. Anal. 5 (2006), no. 4, 855–859. MR 2246012
(2007d:35094), http://dx.doi.org/10.3934/cpaa.2006.5.855
 [MC2]
Li
Ma and Dezhong
Chen, Radial symmetry and monotonicity for an integral
equation, J. Math. Anal. Appl. 342 (2008),
no. 2, 943–949. MR 2445251
(2009m:35151), http://dx.doi.org/10.1016/j.jmaa.2007.12.064
 [MZ]
Li
Ma and Lin
Zhao, Sharp thresholds of blowup and global existence for the
coupled nonlinear Schrödinger system, J. Math. Phys.
49 (2008), no. 6, 062103, 17. MR 2431772
(2009g:35308), http://dx.doi.org/10.1063/1.2939238
 [O]
Biao
Ou, A remark on a singular integral equation, Houston J. Math.
25 (1999), no. 1, 181–184. MR 1675383
(2000e:45004)
 [Se]
James
Serrin, A symmetry problem in potential theory, Arch. Rational
Mech. Anal. 43 (1971), 304–318. MR 0333220
(48 #11545)
 [WX]
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
 [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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
35J99,
45E10,
45G05
Retrieve articles in all journals
with MSC (2010):
35J99,
45E10,
45G05
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
PII:
S 00029939(10)103682
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.
