Article Contents
Article Contents

# Radial symmetry of nonnegative solutions for nonlinear integral systems

• *Corresponding author

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

$$$\left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0<\alpha<n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right.$$$

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.

Mathematics Subject Classification: Primary: 35B06; 35J60; 45G05; Secondary: 45G15.

 Citation:

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