Liouville type theorems for fractional and higher-order fractional H\'enon-Lane-Emden systems

TL;DR

This paper proves non-existence results for fractional and higher-order fractional Hénon-Lane-Emden systems using decay estimates and the method of scaling spheres, and constructs examples of super harmonic functions.

Contribution

It introduces decay estimates and applies the scaling spheres method to establish Liouville type theorems for these systems, extending previous results.

Findings

Decay estimates for fractional systems

Liouville type theorems established

Examples of super harmonic functions constructed

Abstract

In this paper, we first establish decay estimates for the fractional and higher-order fractional H\'enon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in \cite{DQ2} to derive a Liouville type theorem. We also construct an interesting example on super $\frac{\alpha}{2}$-harmonic functions (Proposition 1.2).