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

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## Abstract

In this paper, we are concerned with the weighted elliptic system with the advection term 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.

$$\begin{aligned} {\left\{ \begin{array}{ll} -\omega (x)\Delta u(x)-\nabla \omega (x)\cdot \nabla u(x)=\omega _1 v^{\vartheta },\\ -\omega (x)\Delta v(x)-\nabla \omega (x)\cdot \nabla v(x)=\omega _2 u^p, \end{array}\right. } \quad \text{ in }\;\ \mathbb {R}^N, \end{aligned}$$

### Keywords

Weighted elliptic system Advection term Stable solutions Liouville type theorem### Mathematics Subject Classification

35B33 35B45 35B53 35J60## Notes

### Acknowledgements

The author would wish to express their appreciations to the anonymous referee for his/her valuable suggestions, which have greatly improved this paper. The author wishes to express his warmest thanks to Professor Tianling Jin for his hospitality and Department of Mathematics, Hong Kong University of Science and Technology (where part of this work was done). The work was partially supported by NSFC of China (No. 11471174), Natural Science Foundation of Zhejiang Province (No. LY17A010007) and K.C.Wong Magna Fund in Ningbo University.

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