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Radial symmetry and decay rates of positive solutions of a Wolff type integral system


Authors: Yutian Lei and Chao Ma
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
Published electronically: September 29, 2011
MathSciNet review: 2846322
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Abstract: In this paper, we study the properties of the positive solutions of a nonlinear integral system involving Wolff potentials:

$\displaystyle \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

$\displaystyle 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

DOI: https://doi.org/10.1090/S0002-9939-2011-11401-1
Keywords: Integral equation, Wolff potential, decay rate, radial symmetry
Received by editor(s): November 3, 2010
Published electronically: September 29, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.