Classification of solutions to equations involving Higher-order fractional Laplacian

Abstract

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.