On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms

In this paper, we study the elliptic equations
$$ -G_\alpha u+c({\rm x})\cdot\nabla_\alpha u=h({\rm x} )e^{u}, \qquad {\rm x} = (x,y) \in \mathbb R^{N_{1}}\times \mathbb R^{N_{2}}=\mathbb R^{N}, $$
where $G_{\alpha} =\Delta_{x}+ ( 1+\alpha )^{2}\lvert x\rvert^{2\alpha}\Delta_{y}$, $\alpha > 0$, is the Grushin operator. Here, the advection term $c({\rm x})$ is a smooth, divergence free vector field satisfying certain decay condition and $h({\rm x}) $ is a continuous function such that $h({\rm x} )\geq C|{\rm x}|^l$, $l\geq 0$, where $|{\rm x}|$ is the Grushin norm of ${\rm x}$. We will prove that the equation has no stable solutions provided that
$$ N_{\alpha}< 10+ 4 l, $$
where $N_\alpha:=N_1+(1+\alpha)N_2$ is the homogeneous dimension of $\mathbb R^N$ associated to the Grushin operator.