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In this paper, we study the following high order elliptic equation involvingthe GJMS operator:  \begin{align*}  \alphaP_{\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 newproof of the classical Beckner inequality.

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