Nonlinear Liouville-type theorem for a differential inequality involving Dunkl-Baouendi-Grushin operator.

Let $ R_i $, $ i = 1,2 $, be a root system in $ {\mathbb{R}}^{N_i}\backslash\{0\} $, $ N_i\geq 1 $, $ W_i = W(R_i) $ be the associated finite reflection group, and $ k_i: R_i\to [0,\infty) $ be a multiplicity function, i.e. $ k_i $ is $ W_i $-invariant. For $ \ell\geq 0 $, we introduce the operator $ L_{\ell,k_1,k_2} $ defined by$ L_{\ell,k_1,k_2}u(x,y) = \Delta_{k_1}u(x,y)+|x|^{2\ell}\Delta_{k_2}u(x,y),\,\,(x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}, $where $ \Delta_{k_i} $ is the Dunkl Laplacian operator associated with $ R_i $ and $ k_i $. Our goal in this paper is to establish a Liouville-type result for the semilinear inequality$ -L_{\ell,k_1,k_2}u\geq |u|^p,\,\, (x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}, $where $ u = u(x,y) $ and $ p>1 $. [ABSTRACT FROM AUTHOR]