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

Lei, Yutian; Ma, Chao

doi:10.1090/S0002-9939-2011-11401-1

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