Classification of solutions to equations involving Higher-order fractional Laplacian

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 $p\ge 1$ is an integer, $0<\text{alp}<2$, $n>2p+\alpha$ 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 ${\mathbb{R}}^n$ and monotone decreasing in the radial direction via method of moving planes in integral forms.