Nonexistence of Positive Solutions for high-order Hardy-H$ \acute{e} $non Systems on $ \mathbb{R}^{n} $

In this paper, we study the following high-order Hardy-H$ \acute{e} $non type system:

$ \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 $ 0<\alpha = s_{1}+2<n $, $ 0<\beta = s_{2}+2<n $, $ 0<s_{1},s_{2}<2 $, $ a>-s_{1} $, $ b>-s_{2} $, $ p,q>0 $. There are two cases to be considered. The first one is where the domain is the whole space $ \mathbb{R}^{n} $, and the second one is where the domain is bounded. First of all, we consider the above system in the whole space $ \mathbb{R}^{n} $, we show that the above system are equivalent to the integral system:

$ \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 $ 1<p<\frac{n+\alpha+2a}{n-\alpha} $, $ 1<q<\frac{n+\alpha+2b}{n-\alpha} $ with $ \alpha = \beta $ in the above elliptic system, we prove the nonexistence of a positive solution for the above system in $ \mathbb{R}^{n} $. Then, through the $ Doubling\ Lemma $ we obtain the singularity estimates of the positive solutions on a bounded domain $ \Omega $.