The moving plane method and the uniqueness of high order elliptic equation with GJMS operator

TL;DR

This paper proves the uniqueness of solutions to a high order elliptic equation involving the GJMS operator on spheres, and offers a new proof of the Beckner inequality, expanding understanding of geometric PDEs.

Contribution

It establishes solution uniqueness for certain parameter ranges of the GJMS operator equation and provides a novel proof of the Beckner inequality.

Findings

Solutions are trivial (identically zero) under specified conditions.

New proof of the classical Beckner inequality.

Extends the moving plane method to high order elliptic equations.

Abstract

In this paper, we study the following high order elliptic equation involving the GJMS operator: \begin{align*} \alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}. \end{align*} We establish that if $\alpha>1$ and $n\geq3$, or if $\alpha\in (1-\epsilon_0, 1)$ with $n=2m\geq4$, then $v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.