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

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

Published electronically:
April 14, 2010

MathSciNet review:
2644892

Full-text 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

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. Brazil. Mat. (N.S.)**22**(1) (1991), 1-37. 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), 271-297. MR**982351 (90c:35075)****[CJ]**C. Jin and C. Li,*Symmetry of solutions to some integral equations*, Proc. Amer. Math. Soc.,**134**(2006), 1661-1670. MR**2204277 (2006j:45017)****[CJ1]**C. Jin and C. Li,*Quantitative analysis of some system of integral equations*, Cal. Var. PDEs,**26**(2006), 447-457. MR**2235882 (2007c:45013)****[CL]**W. Chen and C. Li,*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J.,**63**(1991), 615-622. MR**1121147 (93e:35009)****[CL1]**W. Chen and C. Li,*A priori estimates for prescribing scalar curvature equations*, Annals of Math. (2),**145**(1997), 547-564. 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), 1-8. MR**2126275 (2006g:45006)****[CL3]**W. Chen and C. Li,*The best constant in some weighted Hardy-Littlewood-Sobolev inequality.*Proc. Amer. Math. Soc.,**136**(2008), 955-962. 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), 330-343. 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), 347-354. 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), 59-65. MR**2131045 (2006a:45007)****[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. 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 Hardy-Littlewood-Sobolev and related inequalities*, Ann. of Math.,**118**(1983), 349-374. MR**717827 (86i:42010)****[Li]**C. Li,*Local asymptotic symmetry of singular solutions to nonlinear elliptic equations*, Invent. Math.,**123**(1996), 221-231. 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), 1049-1057. 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), 1-12. 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), 855-859. 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), 943-949. MR**2445251 (2009m:35151)****[MZ]**L. Ma and L. Zhao,*Sharp thresholds of blow-up 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), 181-184. MR**1675383 (2000e:45004)****[Se]**J. Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.,**43**(1971), 304-318. MR**0333220 (48:11545)****[WX]**J. Wei and X. Xu,*Classification of solutions of higher order conformally invariant equations*, Math. Ann.,**313**(1999), 207-228. 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:
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.