Journal of Inequalities and Applications (Apr 2020)
On stable entire solutions of sub-elliptic system involving advection terms with negative exponents and weights
Abstract
Abstract We examine the weighted Grushin system involving advection terms given by { Δ G u − a ⋅ ∇ G u = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) v − p in R n , Δ G v − a ⋅ ∇ G v = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) u − q in R n , $$ \textstyle\begin{cases} \Delta _{G} u - a \cdot \nabla _{G} u =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v - a \cdot \nabla _{G} v =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$ where Δ G u = Δ x u + | x | 2 α Δ y u $\Delta _{G} u= \Delta _{x} u+ |x|^{2\alpha } \Delta _{y} u$ , z = ( x , y ) ∈ R n : = R n 1 × R n 2 $\mathbf{z}=(x,y) \in \mathbb {R}^{n}:= \mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}}$ is the Grushin operator, α ≥ 0 $\alpha \geq 0$ , p ≥ q > 1 $p \geq q >1$ , ∥ z ∥ 2 ( α + 1 ) = | x | 2 ( α + 1 ) + | y | 2 $\|\mathbf{z}\|^{2(\alpha +1)}= |x|^{2(\alpha +1)} + |y|^{2} $ , γ ≥ 0 $\gamma \geq 0$ 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 Δ G u − a ⋅ ∇ G u = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) u − p in R n . $$ \Delta _{G} u - a \cdot \nabla _{G} u =\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p} \quad \mbox{in } \mathbb {R}^{n}. $$ 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).
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