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On the classification of solutions to a weighted elliptic system involving the Grushin operator (2007.03009v1)

Published 6 Jul 2020 in math.AP

Abstract: We investigate here the following weighted degenerate elliptic system \begin{align*} -\Delta_{s} u = \Big(1+|\mathbf{x}|{2(s+1)}\Big){\frac{\alpha}{2(s+1)}} vp, \quad -\Delta_{s} v = \Big(1+|\mathbf{x}|{2(s+1)}\Big){\frac{\alpha}{2(s+1)}}u\theta, \quad u,v>0\quad\mbox{in }\; \mathbb{R}N:=\mathbb{R}{N_1}\times \mathbb{R}{N_2}. \end{align*} where $\Delta_{s}=\Delta_{x}+|x|{2s}\Delta_{y},$ is the Grushin operator, $s \geq 0,$ $\alpha \geq 0$ and $1<p\leq\theta.$ Here $$\|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^2\Big)^{\frac{1}{2(s+1)}}, \;\mbox{and}\;\; \mathbf{x}:=(x, y)\in \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.$$ In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results \cite{cow, Hfh, HU, Fa, DP}. As a consequence, we obtain a nonexistence result for the weighted Grushin equation \begin{align*} -\Delta_{s} u =\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} u^p,\;\; \quad u\>0 \quad \mbox{in }\;\; \mathbb{R}N. \end{align*}

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