### A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system

Liang-Gen Hu

#### 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}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $\alpha >-4$, $0 \le \beta < N-4$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and reveal 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 for stable solutions (whether positive or sign-changing) of the weighted elliptic system in the higher dimension.

#### Article information

Source
Adv. Differential Equations Volume 22, Number 1/2 (2017), 49-76.

Dates
First available in Project Euclid: 20 January 2017