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

TL;DR

This paper develops a monotonicity formula for a weighted elliptic system and uses it, along with Pohozaev identity and blow-down analysis, to establish Liouville-type theorems for stable solutions in higher dimensions.

Contribution

It introduces a new monotonicity formula derived from Pohozaev identity and applies it to prove Liouville theorems for stable solutions of weighted elliptic systems.

Findings

Monotonicity formula for solutions of the weighted elliptic system.

Liouville-type theorems for stable solutions in higher dimensions.

Applicability to both positive and sign-changing solutions.

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.