Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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:

$\displaystyle \left \{ \begin {aligned}u_1 &= W_{\beta ,\gamma }(f_1(u)) \\ \vdots \\ u_m &= W_{\beta ,\gamma }(f_m(u)), \end{aligned} \right . $

where $ u=(u_1,\ldots ,u_m)$ and

$\displaystyle W_{\beta ,\gamma }(f)(x)=\int _0^{\infty } [\frac {\int _{B_t(x)}f(y)dy}{t^{n-\beta \gamma }}]^{\frac {1}{\gamma -1}} \frac {dt}{t} $

with $ 1<\gamma <2$ and $ n>\beta \gamma $. First, we estimate the decay rate of the positive solutions at infinity. Based on this, we prove radial symmetry and monotonicity for those solutions by the refined method of moving planes in integral forms, which was established by Chen, Li and Ou. Since the Kelvin transform cannot be used in such a Wolff type system, we have to find a new technique to study the asymptotic estimate, which is essential when we move the planes.

References [Enhancements On Off] (What's this?)

  • 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 $ R^{n}$, 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 $ \gamma $-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 $ n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. MR 0098285 (20:4746)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J50, 45E10, 45G05

Retrieve articles in all journals with MSC (2010): 35J50, 45E10, 45G05

Additional Information

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

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

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.

American Mathematical Society