Abstract:In this paper, we are concerned with the following equations \begin{equation*} \\\begin {cases} (-\Delta )^{m+\frac {\alpha }{2}}u(x)=f(x,u,Du,\cdots ), x\in \mathbb {R}^{n}, \\ u\in C^{2m+[\alpha ],\{\alpha \}+\epsilon }_{loc}\cap \mathcal {L}_{\alpha }(\mathbb {R}^{n}), u(x)\geq 0, x\in \mathbb {R}^{n} \end{cases}\end{equation*} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to the above equations. Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to the above fractional higher-order equations with general nonlinearities $f(x,u,Du,\cdots )$ including conformally invariant and odd order cases. In particular, we classify nonnegative classical solutions to all odd order conformally invariant equations. Our results completely improve the classification results for third order conformally invariant equations in Dai and Qin (Adv. Math., 328 (2018), 822-857) by removing the assumptions on integrability. We also give a crucial characterization for $\alpha$-harmonic functions via outer-spherical averages in the appendix.

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中文翻译：

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超多谐性质、刘维尔定理和高阶分数拉普拉斯算子方程非负解的分类

摘要：在本文中，我们关注以下方程 \begin{equation*} \\\begin {cases} (-\Delta )^{m+\frac {\alpha }{2}}u(x)=f (x,u,Du,\cdots ), x\in \mathbb {R}^{n}, \\ u\in C^{2m+[\alpha ],\{\alpha \}+\epsilon }_{ loc}\cap \mathcal {L}_{\alpha }(\mathbb {R}^{n}), u(x)\geq 0, x\in \mathbb {R}^{n} \end{cases }\end{equation*} 涉及高阶分数拉普拉斯算子。通过引入一种新方法，我们证明了上述方程的非负解的超多谐性质。我们的定理似乎是这个问题的第一个结果。因此，我们推导出了超多谐波特性的许多重要应用。例如，我们建立刘维尔定理，上述具有一般非线性的分数高阶方程的非负解的积分表示公式和分类结果 $f(x,u,Du,\cdots )$ 包括保形不变和奇数阶情况。特别是，我们对所有奇数阶共形不变方程的非负经典解进行分类。我们的结果完全改善了 Dai 和 Qin 中三阶共形不变方程的分类结果（高级数学，328 (2018), 822-857) 通过删除关于可集成性的假设。我们还通过附录中的外球面平均值给出了 $\alpha$-谐波函数的关键特征。

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