American Institute of Mathematical Sciences

Advanced
November  2011, 30(4): 1083-1093. doi: 10.3934/dcds.2011.30.1083

Radial symmetry of solutions for some integral systems of Wolff type

Wenxiong Chen 1, and  Congming Li 2,

1. 

Department of Mathematics, Yeshiva University, New York, NY 10033
2. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  March 2010 Revised  August 2010 Published  May 2011

We consider the fully nonlinear integral systems involving Wolff potentials:

$\u(x) = W_{\beta, \gamma}(v^q)(x)$, $\x \in R^n$;
$\v(x) = W_{\beta, \gamma} (u^p)(x)$, $\x \in R^n$;

(1)

where

$ \W_{\beta,\gamma} (f)(x) = \int_0^{\infty}$ $[ \frac{\int_{B_t(x)} f(y) dy}{t^{n-\beta\gamma}} ]^{\frac{1}{\gamma-1}} \frac{d t}{t}.$

    
   After modifying and refining our techniques on the method of moving planes in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to systems (1).        
   This system includes many known systems as special cases, in particular, when $\beta = \frac{\alpha}{2}$ and $\gamma = 2$, system (1) reduces to

$\u(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} v(y)^q dy$, $\ x \in R^n$,
$v(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} u(y)^p dy$, $\ x \in R^n$.

(2)

The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs

$(-\Delta)^{\alpha/2} u = v^q$, in $R^n$,
$(-\Delta)^{\alpha/2} v = u^p$, in $R^n$

(3)

which comprises the well-known Lane-Emden system and Yamabe equation.
Keywords: radial symmetry, Wolff potentials, nonlinear systems, method of moving planes in integral forms, norm estimates..
Mathematics Subject Classification: Primary: 45G15, 35J60; Secondary: 35J9.
Citation: Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, C. P. A. M., XLII (1989), 271-297.  Google Scholar

[2]

W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (): 164.   Google Scholar

[3]

W. Chen and C. Li, Regularity of Solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8.  Google Scholar

[4]

W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962.  Google Scholar

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 4 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[6]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, submitted to Trans. AMS, 2011. Google Scholar

[9]

W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations," AIMS Book Series on Diff. Equa. & Dyn. Sys., 4, 2010.  Google Scholar

[10]

C. Ma, W. Chen, and C. Li, Regularity of solutions for an integral system of Wolff type, Advances of Math, 3(2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14.  Google Scholar

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[14]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$ "Mathematical Analysis and Applications," Advances in Mathematics, 7a, Academic Press, New York, 1981.  Google Scholar

[15]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res Lett, 2007, preprint, arXiv:math/0703778. Google Scholar

[16]

F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[17]

G. Hardy and J. Littelwood, Some properties of fractional integrals I, Math. Zeitschr., 27 (1928), 565-606. doi: 10.1007/BF01171116.  Google Scholar

[18]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670.  Google Scholar

[19]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[20]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math, 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[22]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 1796-1806.  Google Scholar

[23]

Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.  Google Scholar

[24]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[25]

C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301.  Google Scholar

[26]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925.  Google Scholar

[27]

D. Li and R. Zhuo, An integral equation on half space, Proc. AMS, 138 (2010), 2779-2791.  Google Scholar

[28]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[29]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar

[30]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rat. Mech. Anal., 2 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[31]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Annals of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[32]

S. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.), 4 (1938), 471-479; AMS Transl. Ser. 2, 34 (1963), 39-68. Google Scholar

[33]

N. Trudinger and X. Wang, Hessian measure II, Annals of Math., 150 (1999), 579-604. doi: 10.2307/121089.  Google Scholar

[34]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

show all references

References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, C. P. A. M., XLII (1989), 271-297.  Google Scholar

[2]

W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., 2005 (): 164.   Google Scholar

[3]

W. Chen and C. Li, Regularity of Solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8.  Google Scholar

[4]

W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962.  Google Scholar

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 4 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[6]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, submitted to Trans. AMS, 2011. Google Scholar

[9]

W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations," AIMS Book Series on Diff. Equa. & Dyn. Sys., 4, 2010.  Google Scholar

[10]

C. Ma, W. Chen, and C. Li, Regularity of solutions for an integral system of Wolff type, Advances of Math, 3(2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14.  Google Scholar

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[14]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n,$ "Mathematical Analysis and Applications," Advances in Mathematics, 7a, Academic Press, New York, 1981.  Google Scholar

[15]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res Lett, 2007, preprint, arXiv:math/0703778. Google Scholar

[16]

F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[17]

G. Hardy and J. Littelwood, Some properties of fractional integrals I, Math. Zeitschr., 27 (1928), 565-606. doi: 10.1007/BF01171116.  Google Scholar

[18]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670.  Google Scholar

[19]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[20]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math, 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[22]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 1796-1806.  Google Scholar

[23]

Y. Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.  Google Scholar

[24]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[25]

C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301.  Google Scholar

[26]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925.  Google Scholar

[27]

D. Li and R. Zhuo, An integral equation on half space, Proc. AMS, 138 (2010), 2779-2791.  Google Scholar

[28]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[29]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar

[30]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rat. Mech. Anal., 2 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[31]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Annals of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[32]

S. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.), 4 (1938), 471-479; AMS Transl. Ser. 2, 34 (1963), 39-68. Google Scholar

[33]

N. Trudinger and X. Wang, Hessian measure II, Annals of Math., 150 (1999), 579-604. doi: 10.2307/121089.  Google Scholar

[34]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

[1]

Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
[2]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462
[3]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
[4]

Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
[5]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[6]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
[7]

Huan-Zhen Chen, Zhongxue Lü. Positive solutions to involving Wolff potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 773-788. doi: 10.3934/cpaa.2014.13.773
[8]

Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721
[9]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[10]

Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051
[11]

Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
[12]

Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861
[13]

Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008
[14]

Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385
[15]

Huan Chen, Zhongxue Lü. The properties of positive solutions to an integral system involving Wolff potential. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1879-1904. doi: 10.3934/dcds.2014.34.1879
[16]

JaEun Ku. Maximum norm error estimates for Div least-squares method for Darcy flows. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1305-1318. doi: 10.3934/dcds.2010.26.1305
[17]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[18]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[19]

Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179
[20]

Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (30)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences