Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 161, pp. 1-21.

Title: Regularity and  symmetry of positive solutions to nonlinear
       integral systems

Authors: Wanghe Yao  (Jiangxi Normal Univ., China)
         Xiaoli Chen (Jiangxi Normal Univ., China)
         Jianfu Yang (Jiangxi Normal Univ., China)

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 07, 2011.
Math Subject Classifications: 35J25, 47G30, 35B45, 35J70.
Key Words: L-infinity bounds; Holder continuous;
           radial symmetry; fractional Laplacian