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\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