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 & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
References:
[1]

L. CaffarelliB. 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.  Google Scholar

[2]

W. ChenC. 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.  Google Scholar

[3]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.   Google Scholar

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.   Google Scholar

[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.  Google Scholar

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.   Google Scholar

[7]

C. ChengZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. YinJ. Zhang and X. Shang, The overdetermined equations on bounded domain, Complex Var. Elliptic Equ., 61 (2016), 1566-1586.  doi: 10.1080/17476933.2016.1188296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. ZhuoW. ChenX. 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.   Google Scholar

show all references

References:
[1]

L. CaffarelliB. 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.  Google Scholar

[2]

W. ChenC. 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.  Google Scholar

[3]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.   Google Scholar

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.   Google Scholar

[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.  Google Scholar

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.   Google Scholar

[7]

C. ChengZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. YinJ. Zhang and X. Shang, The overdetermined equations on bounded domain, Complex Var. Elliptic Equ., 61 (2016), 1566-1586.  doi: 10.1080/17476933.2016.1188296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. ZhuoW. ChenX. 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.   Google Scholar

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