Electron. J. Diff. Equ., Vol. 2014 (2014), No. 45, pp. 1-11.

Asymptotic behavior of singular solutions to semilinear fractional elliptic equations

Guowei Lin, Xiongjun Zheng

Abstract:
In this article we study the asymptotic behavior of positive singular solutions to the equation
$$ (-\Delta)^{\alpha} u+u^p=0\quad\text{in } \Omega\setminus\{0\}, $$
subject to the conditions $u=0$ in $\Omega^c$ and $\lim_{x\to0}u(x)=\infty$, where $p\geq1$, $\Omega$ is an open bounded regular domain in $\mathbb{R}^N$ ( $N\ge2$) containing the origin, and $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ denotes the fractional Laplacian. We show that the asymptotic behavior of positive singular solutions is controlled by a radially symmetric solution with $\Omega$ being a ball.

Submitted September 24, 2013. Published February 10, 2014.
Math Subject Classifications: 35R11, 35B06, 35B40.
Key Words: Fractional Laplacian; radial symmetry; asymptotic behavior; singular solution.

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Guowei Lin
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: lgw2008@sina.cn
Xiongjun Zheng
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: xjzh1985@126.com

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