On the nonexistence of stable solutions of sub-elliptic systems with negative exponents

ABSTRACT

In this paper, we study the sub-elliptic system (1) $\begin{array}{rl}{\mathrm{\Delta }}_{\lambda }u& =v^{-p}\\ {\mathrm{\Delta }}_{\lambda }v& =u^{-q}\end{array}\mathrm{in}{\mathbb{R}}^N,$(1) where p,q>1 and ${\mathrm{\Delta }}_{\lambda }$ is an operator of the form ${\mathrm{\Delta }}_{\lambda }=\sum _{i=1}^N{\mathrm{\partial }}_{x_i}\left({\lambda }_i^2{\mathrm{\partial }}_{x_i}\right).$ We establish some Liouville type results for bounded stable positive solutions of the system. As a consequence, we also obtain a nonexistence result for the scalar equation ${\mathrm{\Delta }}_{\lambda }u=u^{-p},\mathrm{in}{\mathbb{R}}^N.$

KEYWORDS:

• Liouville-type result
• stable positive solutions
• degenerate elliptic systems
• negative exponents

AMS SUBJECT CLASSIFICATIONS:

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

Acknowledgment

We would like to express our gratitude to the anonymous referee for the careful reading and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.307.