Abstract:

We study the degenerate elliptic system with weight

$\left\{\begin{array}{ll}-\Delta_{G} u=\omega (x) v, \\-\Delta_{G} v=\omega (x) u^q, \end{array}\right.\quad \mbox{in}\;\ \mathbb{R} ^N=\mathbb{R} ^{N_1}\times \mathbb{R} ^{N_2}, $

where $\Delta_{G} u=\Delta_{x} u+(a+1)^2|x|^{2a}\Delta_{y} u$ is the Grushin operator, $a, \beta\ge0$, $q>1$, $\omega(x)=\left (1+\| x \|^{2(a+1)}\right)^{\frac{\beta}{2(a+1)}}$. Liouville type results for positive stable solutions in the supercritical exponent are established.

Key words: Grushin operator, Stable solution, Liouville theorem, Bootstrap method