Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity

J. Korean Math. Soc. Published online July 29, 2021

Linfen Cao and Xiaoshan Wang
Henan Normal University; Nanjing Normal University

Abstract : 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 $0<s<1$, $\gamma>0$, $1<\beta\leq\frac{n+2s}{n-2s}$, $\lambda>0$ are constants and $a_{0}\geq0$.

We use a direct method of moving planes which introduced by Chen-Li-Li to prove the positive solutions $u(x)$ must be radially symmetric and monotone increasing about some point in $\mathbb{R}^{n}$.