Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity $$ (-\Delta)^{s}u(x)=\lambda u^{\beta}(x)+a_{0}u^{-\gamma}(x), ~ x\in \mathbb{R}^{n}, $$ where $00$, $1<\beta\leq\frac{n+2s}{n-2s}$, $\lambda>0$ are constants and $a_{0}\geq0$. We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions $u(x)$ must be radially symmetric and monotone increasing about some point in $\mathbb{R}^{n}$.

Keywords: Fractional Laplacian, negative powers, method of moving planes

MSC numbers: Primary 35R11; Secondary 35B06

Supported by: The first author is supported by NSFC (No.11671121, 11971153).