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

Let Rᵢ, i = 1, 2, be a root system in R^Nᵢ\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, ₊䃑, ₊䃒 defined by L, ₊䃑, ₊䃒u (x, y) = ₊䃑u (x, y) +|x|^2₊䃒u (x, y), \, \, (x, y) R^N₁ R^N₂, where ₊㶁 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, ₊䃑, ₊䃒u |u|ᵖ, \, \, (x, y) R^N₁ R^N₂, where u = u (x, y) and p>1.