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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.

Monotonicity formula and Liouville-type theorems of stable solution for the weighted elliptic system