Asymptotic behavior of singular solutions to semilinear fractional elliptic equations

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.

Keywords

• Fractional Laplacian
• radial symmetry
• asymptotic behavior
• singular solution