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

Abstract

In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|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 apply Pohozaev identity to construct a monotonicity formula and find their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems of stable solutions for the weighted elliptic system (whether positive or sign-changing) in the higher dimension.