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doi: 10.3934/cpaa.2021201
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Radial symmetry of nonnegative solutions for nonlinear integral systems

Zhenjie Li , and  Chunqin Zhou

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

*Corresponding author

Received  May 2021 Revised  November 2021 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.
Keywords: radial symmetry, integral systems, nonlinear PDE systems, method of moving planes in integral forms, Hardy-Littlewood-Sobolev inequality.
Mathematics Subject Classification: Primary: 35B06; 35J60; 45G05; Secondary: 45G15.
Citation: Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021201
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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.   Google Scholar

[2]

L. Caffarelli, B. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.,  Google Scholar

[7]

D. Li, P. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[12]

R. Zhuo, W. Chen, X. 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.  Google Scholar

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