In this paper, we are concerned with the weighted elliptic system\begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Deltav=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ \Omega,\end{equation*}where $\Omega$ is a subset of $\mathbb{R}^N$, $N \ge 5$, $\alpha>-4$, $0 \le \beta \le \dfrac{N-4}{2}$, $p>1$ and $\vartheta=1$. We first applyPohozaev identity to construct a monotonicity formula and find their certainequivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicityformula} of solutions together with a {\it blowing down} sequence, we proveLiouville-type theorems of stable solutions for the weighted elliptic system(whether positive or sign-changing) in the higher dimension.
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机译：在本文中，我们关注加权椭圆系统\ begin {equation *} \ begin {cases}-\ Delta u = | x | ^ {\ beta} v ^ {\ vartheta}，\\-\\ Deltav = | x | ^ {\ alpha} | u | ^ {p-1} u，\ end {cases} \ quad \ mbox {in} \; \ \ Omega，\ end {equation *}其中$ \ Omega $是子集$ \ mathbb {R} ^ N $，$ N \ ge 5 $，$ \ alpha> -4 $，$ 0 \ le \ beta \ le \ dfrac {N-4} {2} $，$ p> 1 $和$ \ vartheta = 1 $。我们首先应用Pohozaev恒等式构造一个单调公式，并找到它们的确定等价关系。通过使用解的{\ it Pohozaev身份}，{\ it单调性公式}以及{\ it blowdown}序列，我们证明了加权椭圆系统（正向或正负号）稳定解的Liouville型定理在更高的维度。
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