An integral equation on half space

Authors:
Dongyan Li and Ran Zhuo

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2779-2791

MSC (2010):
Primary 35J99, 45E10, 45G05

Published electronically:
April 14, 2010

MathSciNet review:
2644892

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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

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**, 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**, 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**, 10.1090/S0002-9939-05-08411-X**[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**, 10.1007/s00526-006-0013-5**[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**, 10.1215/S0012-7094-91-06325-8**[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**, 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****[CL3]**Wenxiong Chen and Congming Li,*The best constant in a weighted Hardy-Littlewood-Sobolev inequality*, Proc. Amer. Math. Soc.**136**(2008), no. 3, 955–962. MR**2361869**, 10.1090/S0002-9939-07-09232-5**[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**, 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****[CLO2]**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**[CY]**A. Chang and P. Yang,*On uniqueness of an n-th order differential equation in conformal geometry*, Math. Res. Letters,**4**(1997), 1-12.**[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****[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 York-London, 1981, pp. 369–402. MR**634248****[L]**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**[Li]**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**[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**, 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**, 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**, 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**, 10.1016/j.jmaa.2007.12.064**[MZ]**Li Ma and Lin Zhao,*Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system*, J. Math. Phys.**49**(2008), no. 6, 062103, 17. MR**2431772**, 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****[Se]**James Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304–318. MR**0333220****[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**, 10.1007/s002080050258

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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:
https://doi.org/10.1090/S0002-9939-10-10368-2

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.