Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

Foued Mtiri

doi:10.3934/cpaa.2021187

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2022, Volume 21, Issue 2: 541-553. Doi: 10.3934/cpaa.2021187

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Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

Foued Mtiri

Mathematics Department, Faculty of Sciences and Arts, King Khalid University, Muhayil Asir, Saudi Arabia, University of Tunis El Manar, Faculty of Sciences of Tunis Department of Mathematics, Campus University 2092 Tunis, Tunisia

Received: February 28, 2021

Revised: August 31, 2021

Early access: October 2021

Published: February 2022

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Abstract

We examine the following degenerate elliptic system:

$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v>0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!>\!0. $

We prove that the system has no stable solution provided $ p, \theta >0 $ and $ N_s: = N_1+(1+s)N_2< 2 + \alpha + \beta, $ where

$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $

This result is an extension of some results in [15]. In particular, we establish a new integral estimate for $ u $ and $ v $ (see Proposition 1.1), which is crucial to deal with the case $ 0 < p < 1. $

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Mathematics Subject Classification: Primary: 35J60, 35J65; Secondary: 58E05.

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