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March  2022, 21(3): 837-844. doi: 10.3934/cpaa.2021201

Radial symmetry of nonnegative solutions for nonlinear integral systems

School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

*Corresponding author

Received  May 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

Fund Project: The first author is supported by China Postdoctoral Science Foundation No.2021M692085. The second author is partially supported by NSFC Grant 11771285 and 12031012

In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system
$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0<\alpha<n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $
where
$ 0<a_i/2<\alpha $
,
$ f_i(u) $
,
$ 1\leq i\leq m $
, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables
$ u_1 $
,
$ u_2 $
,
$ \cdots $
,
$ u_m $
. By the method of moving planes in integral forms, we show that the nonnegative solution
$ u = (u_1,u_2,\cdots,u_m) $
is radially symmetric when
$ f_i $
satisfies some monotonicity condition.
Citation: Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure and Applied Analysis, 2022, 21 (3) : 837-844. doi: 10.3934/cpaa.2021201
References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. Univ. Math. J., 51 (2002), 37-51. 

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure. Appl. Math., 42 (1989), 271-297.  doi: 10.1002/3160420304.

[3]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie., 29B (2009), 949-960.  doi: 10.1016/S0252-9602(09)60079-5.

[4]

W. Chen and C Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure. Appl. Anna., 12 (2013), 2497-2514.  doi: 10.3934/2013.12.2497.

[5]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. 

[6]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbb{R}^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981.,

[7]

D. LiP. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.  doi: 10.1016/2014.11.029.

[8]

Y. Lv and C. Zhou, Symmetry for an integral system with general nonlinearity, Disc. Cont. Dyna. Syst., 39 (2019), 1533-1543.  doi: 10.3934/dcds.2018121.

[9]

E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbb{R}^n$, Differ. Int. Equ., 9 (1996), 465-479. 

[10]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1996), 369-380. 

[11]

J. Serrin and H. Zou, The existence of positive solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.  doi: 10.1080/03605309808821356.

[12]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. Univ. Math. J., 51 (2002), 37-51. 

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure. Appl. Math., 42 (1989), 271-297.  doi: 10.1002/3160420304.

[3]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie., 29B (2009), 949-960.  doi: 10.1016/S0252-9602(09)60079-5.

[4]

W. Chen and C Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure. Appl. Anna., 12 (2013), 2497-2514.  doi: 10.3934/2013.12.2497.

[5]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. 

[6]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbb{R}^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981.,

[7]

D. LiP. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.  doi: 10.1016/2014.11.029.

[8]

Y. Lv and C. Zhou, Symmetry for an integral system with general nonlinearity, Disc. Cont. Dyna. Syst., 39 (2019), 1533-1543.  doi: 10.3934/dcds.2018121.

[9]

E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbb{R}^n$, Differ. Int. Equ., 9 (1996), 465-479. 

[10]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1996), 369-380. 

[11]

J. Serrin and H. Zou, The existence of positive solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.  doi: 10.1080/03605309808821356.

[12]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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