# Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

## TL;DR

This paper introduces a new method to establish super poly-harmonic properties for nonnegative solutions of higher-order fractional Laplacian equations, leading to Liouville theorems and classification results that improve previous findings.

## Contribution

The paper presents the first proof of super poly-harmonic properties for these equations and extends classification results without integrability assumptions.

## Key findings

- Established super poly-harmonic properties for solutions
- Derived Liouville theorems and classification results
- Improved previous results by removing integrability constraints

## Abstract

In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlinearities $f(x,u,Du,\cdots)$ including conformally invariant and odd order cases. In particular, our results completely improve the classification results for third order equations in Dai and Qin \cite{DQ1} by removing the assumptions on integrability. We also derive a characterization for $\alpha$-harmonic functions via averages in the appendix.