Liouville Theorems for Stable Weak Solutions of Elliptic Problems Involving Grushin Operator

We consider the boundary value problem

where Ω is a bounded or unbounded C¹ domain of  are nonnegative functions, f is an increasing function, ∇_G and div_G are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on  and f. We also show the sharpness of our results when Ω =  and f has power or exponential growth.