On stable entire solutions of sub-elliptic system involving advection terms with negative exponents and weights

[摘要] We examine the weighted Grushin system involving advection terms given by $$ \textstyle\begin{cases} \Delta _{G} u - a \cdot \nabla _{G} u =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v - a \cdot \nabla _{G} v =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$ where $\Delta _{G} u= \Delta _{x} u+ |x|^{2\alpha } \Delta _{y} u$, $\mathbf{z}=(x,y) \in \mathbb {R}^{n}:= \mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}}$ is the Grushin operator, $\alpha \geq 0$, $p \geq q >1$, $\|\mathbf{z}\|^{2(\alpha +1)}= |x|^{2(\alpha +1)} + |y|^{2} $, $\gamma \geq 0$ and a is a smooth divergence-free vector that we will specify later. Inspired by recent progress in the study of the Lane–Emden system, we establish some Liouville-type results for bounded stable positive solutions of the system. In particular, we prove the comparison principle to establish our result. As consequences, we obtain a Liouville-type theorem for the weighted Grushin equation involving advection terms $$ \Delta _{G} u - a \cdot \nabla _{G} u =\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p} \quad \mbox{in } \mathbb {R}^{n}. $$ The main tools in the proof of the main result are the comparison principle, nonlinear integral estimates via the stability assumption and the bootstrap argument. Our results generalize and improve the previous work in (Duong et al. in Complex Var. Elliptic Equ. 64(12):2117–2129, 2019).

[学科分类] 电力

[关键词] Liouville-type theorem;Grushin operator;Stable solutions;Elliptic system;Weighted Grushin system;Weighted Grushin equation