Let $0<\alpha<2$, $p\geq 1$, $m\in\mathbb{ N}_+$. Consider $u$ to be the positive solution of the PDE \begin{ equation}\label{ abstract PDE} (-\Delta)^{ \frac{ \alpha}{ 2}+m} u(x)=u^p(x) \quad\text{ in }\mathbb{ R}^n. \end{ equation} Cao, Dai and Qin( Transactions of the American mathematical society, 2021) showed that, under the condition $u\in\mathcal{ L}_\alpha$, the PDE possesses super polyharmonic property $(-\Delta)^{ k+\frac{ \alpha}{ 2}}u\geq 0$ for $k=0,1,...,m-1$. In this paper, we show another kind of super polyharmonic property $(-\Delta)^k u> 0$ for $k=1,...m$ under different conditions $(-\Delta)^mu\in\mathcal{ L}_\alpha$ and $(-\Delta)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to the equivalence between the PDE and the integral equation $u(x)=\int_{ \mathbb{ R}^n}\frac{ u^p(y)}{ |x-y|^{ n-2m-\alpha}}dy$.
Categories: math.AP