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Radial symmetry of nonnegative solutions for nonlinear integral systems

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

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  • 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.

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


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