In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{ equation*} \left\{ \begin{ aligned} &(-\Delta)^{ p+{ \frac{ \alpha}{ 2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{ R}^n,\\ &\int_{ \mathbb{ R}^n}u_+^\gamma dx<+\infty, \end{ aligned}\right. \end{ equation*} where $p\geq 1$ is an integer, $0<\alp<2$, $n> 2p+\alpha$ and $\gamma \in (1,\frac{ n}{ n-2p-\alp})$. We establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this we prove that these solutions are radially symmetric about some point in $\R^n$ and monotone decreasing in the radial direction via method of moving planes in integral forms.
Categories: math.AP