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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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 system
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by Yutian Lei and Chao Ma PDF
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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Additional Information
  • 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
  • Email: chao.ma@colorado.edu
  • Received by editor(s): November 3, 2010
  • Published electronically: September 29, 2011
  • Communicated by: Matthew J. Gursky
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 541-551
  • MSC (2010): Primary 35J50, 45E10, 45G05
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11401-1
  • MathSciNet review: 2846322