Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On super polyharmonic property of high-order fractional Laplacian   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
$u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-\alpha}}dy$.

31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an