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

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.