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

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  • 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. $

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


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