# American Institute of Mathematical Sciences

March  2019, 39(3): 1533-1543. 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 and Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. 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.

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##### 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.
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