Liouville-type theorems for sub-elliptic systems involving Δ_λ-Laplacian

Abstract

We study the Liouville-type theorems for the sub-elliptic inequality $-{\mathrm{\Delta }}_{\lambda }u\ge u^p\mathrm{i}\mathrm{n}{\mathbb{R}}^N$ and for the corresponding system $\begin{array}{l}-{\mathrm{\Delta }}_{\lambda }u\ge v^p\\ -{\mathrm{\Delta }}_{\lambda }v\ge u^q\end{array}\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$ where $p,q\in \mathbb{R}$, and ${\mathrm{\Delta }}_{\lambda }$ is a strongly degenerate operator of the form ${\mathrm{\Delta }}_{\lambda }=\sum _{i=1}^N{\mathrm{\partial }}_{x_i}\left({\lambda }_i^2{\mathrm{\partial }}_{x_i}\right).$ Here the functions ${\lambda }_i,i=1,2,\dots ,N,$ satisfy certain conditions such that ${\mathrm{\Delta }}_{\lambda }$ is homogeneous of degree 2 with respect to a group of dilations in ${\mathbb{R}}^N$.

We first prove that the scalar inequality has no positive solution provided $-\mathrm{\infty }<p\le \frac{Q}{Q-2},$ where Q is the homogeneous dimension of ${\mathbb{R}}^N$ associated to the operator ${\mathrm{\Delta }}_{\lambda }$.

Then, we establish the nonexistence of positive solutions of the corresponding system in one of following cases:

1. $p\le 0$ or $q\le 0$,

2. p, q>0 and $pq\le 1$,

3. p, q>0, pq>1 and $max\left\{\frac{2(p+1)}{pq-1},\frac{2(q+1)}{pq-1}\right\}>Q-2$.

Our proofs are based on a maximum principle argument combined with a refinement of Souto's reduction.

Keywords:

• Liouville-type theorems
• ${\mathrm{\Delta }}_{\lambda }$-Laplacian
• system of degenerate elliptic inequalities

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35B53
• 35J60
• Secondary: 35B35
• 35J70

Acknowledgements

The authors would like to thank the anonymous referee for the careful reading and helpful suggestions which improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work is funded by Vietnam Ministry of Education and Training under grant number B2019-SPH-02.