Electron. J. Diff. Equ., Vol. 2011 (2011), No. 161, pp. 1-21.

Regularity and symmetry of positive solutions to nonlinear integral systems

Wanghe Yao, Xiaoli Chen, Jianfu Yang

Abstract:
In this article, we consider the regularity and symmetry of positive solutions to the nonlinear integral system
$$ u(x)=\int_{\mathbb{R}^n}K_{\alpha}(x-y)\frac{v(y)^q}{|y|^\beta}\,dy, \quad v(x)=\int_{\mathbb{R}^n}K_{\alpha}(x-y)\frac{u(y)^p}{|y|^\beta}\,dy $$
for $x\in \mathbb{R}^n$, where $K_\alpha(x)$ is the kernel of the operator $(- \Delta)^{\alpha}+ id$ of order $\alpha$, with $0\leq \beta<2\alpha<n$, $1<p$, $q<(n-\beta)/\beta$ and
$$ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2\alpha+\beta}{n}. $$
We show that positive solution pairs $(u,v)\in L^{p+1}(\mathbb{R}^n)\times L^{q+1}(\mathbb{R}^n)$ are locally Holder continuous in $\mathbb{R}^N\setminus\{0\}$, radially symmetric about the origin, and strictly decreasing.

Submitted July 9, 2011. Published December 7, 2011.
Math Subject Classifications: 35J25, 47G30, 35B45, 35J70.
Key Words: L-infinity bounds; Holder continuous; radial symmetry; fractional Laplacian.

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Wanghe Yao
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: yaowanghe198610@sina.com
Xiaoli Chen
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: littleli_chen@163.com
Jianfu Yang
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jfyang_2000@yahoo.com

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