# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Radial symmetry and decay rates of positive solutions of a Wolff type integral systemHTML articles powered by AMS MathViewer

by Yutian Lei and Chao Ma
Proc. Amer. Math. Soc. 140 (2012), 541-551 Request permission

## Abstract:

In this paper, we study the properties of the positive solutions of a nonlinear integral system involving Wolff potentials: \left \{ \begin {aligned} u_1 &= W_{\beta ,\gamma }(f_1(u)) \\ \vdots \\ u_m &= W_{\beta ,\gamma }(f_m(u)), \end {aligned} \right . where $u=(u_1,\ldots ,u_m)$ and $W_{\beta ,\gamma }(f)(x)=\int _0^{\infty } [\frac {\int _{B_t(x)}f(y)dy}{t^{n-\beta \gamma }}]^{\frac {1}{\gamma -1}} \frac {dt}{t}$ with $1<\gamma <2$ and $n>\beta \gamma$. First, we estimate the decay rate of the positive solutions at infinity. Based on this, we prove radial symmetry and monotonicity for those solutions by the refined method of moving planes in integral forms, which was established by Chen, Li and Ou. Since the Kelvin transform cannot be used in such a Wolff type system, we have to find a new technique to study the asymptotic estimate, which is essential when we move the planes.
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• Yutian Lei
• Affiliation: Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, People’s Republic of China
• Email: leiyutian@njnu.edu.cn
• Chao Ma
• Affiliation: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309