In this paper, we study the following high-order Hardy-H
$ \begin{cases} \ (-\Delta)^{\frac{\alpha}{2}}u(x) = |x|^{a}v^{p}(x) ,\\ \ (-\Delta)^{\frac{\beta}{2}}v(x) = |x|^{b}u^{q}(x) ,\\ \end{cases} $
where
$ \begin{cases} \ u(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{a}v^{p}(y)}{|x-y|^{n-\alpha}}dy,\\[1.5mm] \ v(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{b}u^{q}(y)}{|x-y|^{n-\beta}}dy.\\ \end{cases} $
Then by using the method of moving planes in integral forms, we prove that there are no positive solutions for the above integral system. In addition, while in the subcritical case
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