Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system

Shiren Zhu; Xiaoli Chen; Jianfu Yang

doi:10.3934/cpaa.2013.12.2685

Article Contents

2013, Volume 12, Issue 6: 2685-2696. Doi: 10.3934/cpaa.2013.12.2685

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Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system

1.
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022

Received: August 31, 2012

Revised: November 30, 2012

Published: October 2013

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Abstract

In this paper, we are concerned with properties of positive solutions of the following fractional elliptic system \begin{eqnarray} {(-\Delta+I)}^{\frac{\alpha}{2}}u=\frac{u^pv^q}{|x|^\beta}, \quad {(-\Delta+I)}^{\frac{\alpha}{2}}v=\frac{v^pu^q}{|x|^\beta}\quad in\quad R^n, \end{eqnarray} where $n \geq 3$, $0 \le \beta < \alpha < n$, $ p, q>1$ and $p+q<\frac{n+\alpha-\beta}{n-\alpha+\beta}$. We show that positive solutions of the system are radially symmetric and belong to $L^\infty(R^n)$, which possibly implies that the solutions are locally Hölder continuous. Moreover, if $ \alpha=2, \beta =0,p\le q$, we show that positive solution pair $(u,v)$ of the system is unique and $u=v = U$, where $U$ is the unique positive solution of the problem \begin{eqnarray} -\Delta u + u = u^{p+q}\quad {\rm in}\quad \mathbb{R}^n. \end{eqnarray}

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Mathematics Subject Classification: Primary: 35J25, 47G30, 35B453, 35J70.

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