Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2685 - 2696, Volume 12, Issue 6, November 2013      doi:10.3934/cpaa.2013.12.2685

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Shiren Zhu - Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China (email)
Xiaoli Chen - Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China (email)
Jianfu Yang - Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China (email)

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 HÃ¶lder 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}

Keywords:  Regularity, radially symmetry, uniqueness, Bessel kernel.
Mathematics Subject Classification:  Primary: 35J25, 47G30, 35B453, 35J70.

Received: September 2012;      Revised: December 2012;      Available Online: May 2013.

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