Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: $$ \varvec{p \ge \vartheta >1}$$

Abstract

In this paper, we are concerned with the weighted elliptic system with the advection term $$\begin{aligned} {\left\{ \begin{array}{ll} -\omega (x)\Delta u(x)-\nabla \omega (x)\cdot \nabla u(x)=\omega _1 v^{\vartheta }, -\omega (x)\Delta v(x)-\nabla \omega (x)\cdot \nabla v(x)=\omega _2 u^p, \end{array}\right. } \quad \text{ in }\; \mathbb {R}^N, \end{aligned}$$ where $$N \ge 3$$ , $$p \ge \vartheta >1$$ and $$\omega , \omega _1, \omega _2 \ne 1$$ satisfy some suitable conditions. We establish Liouville type theorems for stable solutions with two cases $$\omega _1 \ne \omega _2$$ and $$\omega _1 \equiv \omega _2$$ , respectively. Based on a delicate application of some new techniques, these difficulties caused by the advection term and the weighted term are overcome, and the sharp results are obtained.