A Liouville theorem for high order degenerate elliptic equations

TL;DR

This paper establishes a Liouville theorem for high order degenerate elliptic equations using the moving plane method, classifying solutions in critical and subcritical cases.

Contribution

It extends Liouville theorems to high order degenerate elliptic equations with a new classification of solutions.

Findings

Liouville theorem for subcritical case

Classification of solutions in critical case

Application of moving plane method to degenerate equations

Abstract

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator $A=y\partial_y^2+a\partial_y+\Delta_x,a\geq 1$. We get a Liouville theorem for subcritical case and classify the solutions for the critical case.