Classification of solutions to equations involving Higher-order fractional Laplacian

TL;DR

This paper classifies solutions to a class of higher-order fractional Laplacian equations, proving symmetry and monotonicity properties of solutions using integral representation and the method of moving planes.

Contribution

It introduces an integral representation formula for solutions and establishes their radial symmetry and monotonicity, extending understanding of fractional Laplacian equations.

Findings

Solutions are radially symmetric about some point.

Solutions are monotone decreasing in the radial direction.

Integral representation formula for solutions is established.

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.