Radial symmetry and decay rates of positive solutions of a Wolff type integral system

Authors:
Yutian Lei and Chao Ma

Journal:
Proc. Amer. Math. Soc. **140** (2012), 541-551

MSC (2010):
Primary 35J50, 45E10, 45G05

DOI:
https://doi.org/10.1090/S0002-9939-2011-11401-1

Published electronically:
September 29, 2011

MathSciNet review:
2846322

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the properties of the positive solutions of a nonlinear integral system involving Wolff potentials:

**1.**L. Caffarelli, B. Gidas, and J. Spruck,*Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math.,**42**(1989), 271-297. MR**982351 (90c:35075)****2.**W. Chen and C. Li,*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J.,**63**(1991), 615-622. MR**1121147 (93e:35009)****3.**W. Chen and C. Li,*A priori estimates for prescribing scalar curvature equations*, Ann. of Math. (2),**145**(1997), 547-564. MR**1454703 (98d:53049)****4.**W. Chen and C. Li,*The best constant in a weighted Hardy-Littlewood-Sobolev inequality*, Proc. Amer. Math. Soc.,**136**(2008), 955-962. MR**2361869 (2009b:35098)****5.**W. Chen and C. Li,*Radial symmetry of solutions for some integral systems of Wolff type*, Disc. Cont. Dynamics Sys.,**30**(2011), 1083-1093.**6.**W. Chen and C. Li,*Classification of positive solutions for nonlinear differential and integral systems with critical exponents*, Acta Mathematica Scientia,**29B**(2009), 949-960. MR**2510000 (2010i:35078)****7.**W. Chen, C. Li, and B. Ou,*Classification of solutions for a system of integral equations*, Comm. in Partial Differential Equations,**30**(2005), 59-65. MR**2131045 (2006a:45007)****8.**W. Chen, C. Li, and B. Ou,*Classification of solutions for an integral equation*, Comm. Pure and Appl. Math.,**59**(2006), 330-343. MR**2200258 (2006m:45007a)****9.**C. Cascante, J. Ortega, and I. Verbitsky,*Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities*, Potential Analysis,**16**(2002), 347-372. MR**1894503 (2003f:31006a)****10.**S-Y. A. Chang and P. Yang,*On uniqueness of an n-th order differential equation in conformal geometry*, Math. Res. Letters,**4**(1997), 91-102. MR**1432813 (97m:58204)****11.**L. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, Cambridge, 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*Mathematical Analysis and Applications*, vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. MR**634248 (84a:35083)****13.**L.I. Hedberg and T. Wolff,*Thin sets in nonlinear potential theory,*Ann. Inst. Fourier (Grenoble),**33**(1983), 161-187. MR**727526 (85f:31015)****14.**C. Jin and C. Li,*Symmetry of solutions to some systems of integral equations*, Proc. Amer. Math. Soc.,**134**(2006), 1661-1670. MR**2204277 (2006j:45017)****15.**C. Jin and C. Li,*Qualitative analysis of some systems of integral equations*, Calc. Var. PDEs,**26**(2006), 447-457. MR**2235882 (2007c:45013)****16.**T. Kilpelaiinen and J. Maly,*Degenerate elliptic equations with measure data and nonlinear potentials*, Ann. Scuola Norm. Sup. Pisa, Cl. Sci.,**19**(1992), 591-613. MR**1205885 (94c:35091)****17.**T. Kilpelaiinen and J. Maly,*The Wiener test and potential estimates for quasilinear elliptic equations*, Acta Math.,**172**(1994), 137-161. MR**1264000 (95a:35050)****18.**D. Labutin,*Potential estimates for a class of fully nonlinear elliptic equations*, Duke Math. J.,**111**(2002), 1-49. MR**1876440 (2002m:35053)****19.**Y. Lei, C. Li, and C. Ma,*Decay estimation for positive solutions of a -Laplace equation*, Discrete Contin. Dyn. Syst.,**30**(2011), 547-558. MR**2772129****20.**Y. Lei and C. Ma,*Asymptotic behavior for solutions of some integral equations*, Comm. Pure Appl. Anal.**10**(2011), 193-207. MR**2746534****21.**C. Li,*Local asymptotic symmetry of singular solutions to nonlinear elliptic equations*, Invent. Math.,**123**(1996), 221-231. MR**1374197 (96m:35085)****22.**C. Li and J. Lim,*The singularity analysis of solutions to some integral equations,*Comm. Pure Appl. Anal.,**6**(2007), 453-464. MR**2289831 (2008e:45008)****23.**C. Li and L. Ma,*Uniqueness of positive bound states to Schrödinger systems with critical exponents,*SIAM J. Math. Anal.,**40**(2008), 1049-1057. MR**2452879 (2009k:35079)****24.**Y. Li,*Remark on some conformally invariant integral equations: the method of moving spheres*, Journal of European Mathematical Society,**6**(2004), 153-180. MR**2055032 (2005e:45007)****25.**E. Lieb,*Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities*, Ann. of Math. (2)**118**(1983), 349-374. MR**717827 (86i:42010)****26.**C. Ma, W. Chen, and C. Li,*Regularity of solutions for an integral system of Wolff type,*Adv. Math.,**226**(2011), 2676-2699. MR**2739789****27.**J. Maly,*Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals*, Manuscripta Math.,**110**(2003), 513-525. MR**1975101 (2004i:35073)****28.**N. Phuc and I. Verbitsky,*Quasilinear and Hessian equations of Lane-Emden type*, Ann. of Math. (2)**168**(2008), 859-914. MR**2456885 (2010a:35075)****29.**J. Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.,**43**(1971), 304-318. MR**0333220 (48:11545)****30.**J. Serrin and H. Zou,*Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities*, Acta Math.,**189**(2002), 79-142. MR**1946918 (2003j:35107)****31.**E. M. Stein and G. Weiss,*Fractional integrals in -dimensional Euclidean space*, J. Math. Mech.,**7**(1958), 503-514. MR**0098285 (20:4746)**

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Additional Information

**Yutian Lei**

Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, People’s Republic of China

Email:
leiyutian@njnu.edu.cn

**Chao Ma**

Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309

Email:
chao.ma@colorado.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11401-1

Keywords:
Integral equation,
Wolff potential,
decay rate,
radial symmetry

Received by editor(s):
November 3, 2010

Published electronically:
September 29, 2011

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.