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

Abstract

<p style='text-indent:20px;'>In this paper, we are concerned with the following high order degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE111000">\begin{document}$\left\{ \begin{align} &amp; {{(-A)}^{m}}u={{v}^{p}} \\ &amp; {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y&gt;0 \right\}, \\ &amp; u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where the operator <inline-formula><tex-math id="M1">\begin{document}$ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ n+2a&gt;2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $\end{document}</tex-math></inline-formula>. We prove the non-existence of positive smooth solutions for <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p,\, q&lt;\frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>, and classify positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p = q = \frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{p+1}+\frac{1}{q+1}&gt;\frac{n+2a-2m}{n+2a} $\end{document}</tex-math></inline-formula>, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities <inline-formula><tex-math id="M6">\begin{document}$ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}_+^{n+1} $\end{document}</tex-math></inline-formula> for either <inline-formula><tex-math id="M8">\begin{document}$ (n+2a-2m)q&lt;\frac{n+2a}{p}+2m $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M9">\begin{document}$ (n+2a-2m)p&lt;\frac{n+2a}{q}+2m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M10">\begin{document}$ p,q&gt;1 $\end{document}</tex-math></inline-formula>.</p>