# American Institute of Mathematical Sciences

• Previous Article
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians
• DCDS Home
• This Issue
• Next Article
On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints
doi: 10.3934/dcds.2018121

## Symmetry for an integral system with general nonlinearity

 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Yingshu Lü

Received  July 2017 Revised  December 2017 Published  April 2018

In this paper, we study the radial symmetry of the solution to the following system of integral form:
 $\left\{ {\begin{array}{*{20}{l}}{{u_i}(x) = \int_{{{\bf{R}}^n}} {\frac{1}{{|x - y{|^{n - \alpha }}}}} {f_i}(u(y))dy,\;\;x \in {{\bf{R}}^n},\;\;i = 1, \cdots ,m,}\\{0 < \alpha < n,\;{\rm{ and }}\;u(x) = ({u_1}(x),{u_2}(x), \cdots ,{u_m}(x)).}\end{array}} \right.\;\;\;\;\;\;\left( 1 \right)$
Here
 $f_i(s)∈ C^1(\mathbf{R^m_+})\bigcap$
 $C^0(\mathbf{\overline{R^m_+}})$
 $(i = 1,2,···,m)$
are real-valued functions, nonnegative and monotone nondecreasing with respect to the variables
 $s_1$
,
 $s_2$
,
 $···$
,
 $s_m$
. We show that the nonnegative solution
 $u = (u_1,u_2,···,u_m)$
is radially symmetric in the general condition that
 $f_i$
satisfies monotonicity condition which contains the critical and subcritical homogeneous degree as special cases. The main technique we use is the method of moving planes in an integral form. Due to our condition here is more general, the more subtle method is needed to deal with this difficulty.
Citation: Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2018121
##### References:
 [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. doi: 10.1002/cpa.3160420304. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. [5] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critial exponents, Acta Math. Sci., 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [6] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. [7] C. Cheng, Z. Lü and Y. Lü, A direct method of moving planes for the system of the fractional Laplacian, Pacific J. Math., 290 (2017), 301-320. doi: 10.2140/pjm.2017.290.301. [8] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [9] C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [10] Y. Lei and C. Ma, Radial symmetry and decay rates of positive solutions of a Wolff type integral system, Proc.Amer.Math.Soc., 140 (2012), 541-551. doi: 10.1090/S0002-9939-2011-11401-1. [11] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [12] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [13] C. Lin and J. V. Prajapat, Asymptotic symmetry of singular solutions of semilinear elliptic equations, J. Differential Equations., 245 (2008), 2534-2550. doi: 10.1016/j.jde.2008.01.022. [14] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian system, Nonlinear Anal., 146 (2016), 120-135. doi: 10.1016/j.na.2016.08.022. [15] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal., 8 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [16] R. Yin, J. Zhang and X. Shang, The overdetermined equations on bounded domain, Complex Var. Elliptic Equ., 61 (2016), 1566-1586. doi: 10.1080/17476933.2016.1188296. [17] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. [18] X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity, J.Differential Equations., 254 (2013), 2173-2182. doi: 10.1016/j.jde.2012.11.021. [19] X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity, Discrete Contin. Dyn. Syst., 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947. [20] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.

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, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. [5] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critial exponents, Acta Math. Sci., 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [6] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. [7] C. Cheng, Z. Lü and Y. Lü, A direct method of moving planes for the system of the fractional Laplacian, Pacific J. Math., 290 (2017), 301-320. doi: 10.2140/pjm.2017.290.301. [8] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [9] C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [10] Y. Lei and C. Ma, Radial symmetry and decay rates of positive solutions of a Wolff type integral system, Proc.Amer.Math.Soc., 140 (2012), 541-551. doi: 10.1090/S0002-9939-2011-11401-1. [11] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. [12] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [13] C. Lin and J. V. Prajapat, Asymptotic symmetry of singular solutions of semilinear elliptic equations, J. Differential Equations., 245 (2008), 2534-2550. doi: 10.1016/j.jde.2008.01.022. [14] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian system, Nonlinear Anal., 146 (2016), 120-135. doi: 10.1016/j.na.2016.08.022. [15] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal., 8 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [16] R. Yin, J. Zhang and X. Shang, The overdetermined equations on bounded domain, Complex Var. Elliptic Equ., 61 (2016), 1566-1586. doi: 10.1080/17476933.2016.1188296. [17] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. [18] X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity, J.Differential Equations., 254 (2013), 2173-2182. doi: 10.1016/j.jde.2012.11.021. [19] X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity, Discrete Contin. Dyn. Syst., 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947. [20] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
 [1] 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 [2] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [3] 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 [4] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [5] Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041 [6] 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 [7] 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 [8] Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475 [9] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [10] Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1 [11] Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056 [12] Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007 [13] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [14] Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167 [15] Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-17. doi: 10.3934/dcdsb.2018204 [16] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004 [17] Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1 [18] Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435 [19] Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032 [20] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

2017 Impact Factor: 1.179