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

In this paper, we study the following high order elliptic equation involving the GJMS operator: \begin{align*} αP_{$\mathbb{S}$^n}v_α+2Q_{g_{$\mathbb{S}$^n}}=2Q_{g_{$\mathbb{S}$^n}}e^{nv_α}. \end{align*} We establish that if $\alpha >1$ and $n\ge 3$, or if $\alpha \in (1-{\epsilon }_0,1)$ with $n=2m\ge 4$, then $v_{\alpha }\equiv 0$. As an application, we present a new proof of the classical Beckner inequality.