Liouville-type theorem for high order degenerate Lane-Emden system

In this paper, we are concerned with the following high order degenerate elliptic system:

$\left\{ \begin{align} & {{(-A)}^{m}}u={{v}^{p}} \\ & {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y>0 \right\}, \\ & u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where the operator $ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $ and $ n+2a>2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $. We prove the non-existence of positive smooth solutions for $ 1<p,\, q<\frac{n+2a+2m}{n+2a-2m} $, and classify positive solutions for $ p = q = \frac{n+2a+2m}{n+2a-2m} $. For $ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n+2a-2m}{n+2a} $, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities $ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $ in $ \mathbb{R}_+^{n+1} $ for either $ (n+2a-2m)q<\frac{n+2a}{p}+2m $ or $ (n+2a-2m)p<\frac{n+2a}{q}+2m $ with $ p,q>1 $.