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

Abstract

<p style='text-indent:20px;'>In this paper, we study the following high-order Hardy-H<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>non type system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M4">\begin{document}$ 0&lt;\alpha = s_{1}+2&lt;n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\beta = s_{2}+2&lt;n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 0&lt;s_{1},s_{2}&lt;2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ a&gt;-s_{1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ b&gt;-s_{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ p,q&gt;0 $\end{document}</tex-math></inline-formula>. There are two cases to be considered. The first one is where the domain is the whole space <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, and the second one is where the domain is bounded. First of all, we consider the above system in the whole space <inline-formula><tex-math id="M11">\begin{document}$ \mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, we show that the above system are equivalent to the integral system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Then by using the method of moving planes in integral forms, we prove that there are no positive solutions for the above integral system. In addition, while in the subcritical case <inline-formula><tex-math id="M12">\begin{document}$ 1&lt;p&lt;\frac{n+\alpha+2a}{n-\alpha} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ 1&lt;q&lt;\frac{n+\alpha+2b}{n-\alpha} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ \alpha = \beta $\end{document}</tex-math></inline-formula> in the above elliptic system, we prove the nonexistence of a positive solution for the above system in <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{R}^{n} $\end{document}</tex-math></inline-formula>. Then, through the <inline-formula><tex-math id="M16">\begin{document}$ Doubling\ Lemma $\end{document}</tex-math></inline-formula> we obtain the singularity estimates of the positive solutions on a bounded domain <inline-formula><tex-math id="M17">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>.</p>