关键词: super polyharmonic, fractional Laplacian, equivalence, classification

Abstract: Let $0<\alpha<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider the positive solution $u$ of the PDE
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (-\Delta)^{\frac{\alpha}{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0.1) $
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition $u\in\mathcal{L}_\alpha$, (0.1) possesses a super polyharmonic property $(-\Delta)^{k+\frac{\alpha}{2}}u\geq 0$ for $k=0,1,\cdots ,m-1$. In this paper, we show another kind of super polyharmonic property $(-\Delta)^k u> 0$ for $k=1,\cdots ,m-1$, under the conditions $(-\Delta)^mu\in\mathcal{L}_\alpha$ and $(-\Delta)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation $u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-\alpha}}{\rm d}y$. One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.

Key words: super polyharmonic, fractional Laplacian, equivalence, classification