Liouville-type results for nonlinear elliptic equations with Baouendi-Grushin operators and nonlinear gradient terms

We study positive super-solutions of the nonlinear elliptic equation $$ -\Delta_\gamma u + \frac{b}{u^{\beta-1}} |\nabla_\gamma u|^\beta = c u \quad \text{in } \Omega, $$ with $0 < \beta \le 2$, where $\Omega$ is either a bounded domain or an unbounded exterior domain in $\mathbb{R}^N$, and $\Delta_\gamma$ is the Baouendi--Grushin operator. Extending classical Liouville-type results known for the Laplacian ($\gamma=0$, $\beta=1$) to the sub-elliptic Grushin setting, we derive conditions on the coefficient functions $b$ and $c$ ensuring the absence or existence of positive super-solutions, and provide explicit examples illustrating the sharpness of our results. Our approach employs generalized Hardy inequalities adapted to the Grushin structure, allowing treatment of degenerate and unbounded coefficients without restrictive boundedness assumptions.