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

Let Rᵢ, $ i = 1,2 $, be a root system in RNᵢ\0\, Nᵢ≥ 1, Wᵢ = W(Rᵢ) be the associated finite reflection group, and kᵢ: Rᵢ→ [0,∞) be a multiplicity function, i.e. kᵢ is Wᵢ-invariant. For ≥ 0, we introduce the operator L,k₁,k₂ defined by L,k₁,k₂u(x,y) = Δk₁u(x,y)+|x|²Δk₂u(x,y),\,\,(x,y)∈ RN₁× RN₂, where Δkᵢ is the Dunkl Laplacian operator associated with Rᵢ and kᵢ. Our goal in this paper is to establish a Liouville-type result for the semilinear inequality-L,k₁,k₂u≥ |u|ᵖ,\,\, (x,y)∈ RN₁× RN₂,where $ u = u(x,y) $ and $ p>1 $.