NONEXISTENCE RESULT FOR QUASILINEAR ELLIPTIC INEQUALITIES INVOLVING THE GRUSHIN OPERATOR

In this paper, we establish a nonexistence result of positive solutions of the inequality \( -{\rm div}_G \left( |\nabla_G u|^{p-2} \nabla_G u \right) \geq u^q \quad {\rm in} \; \mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \) where \( \nabla_G = (\nabla_x, |x|^a \nabla_y) \) is the Grushin gradient, \( p, q > 1 \). Here \( (x, y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} \) and \( a > 0 \). As a generalization of this result, we also establish a nonexistence result of positive solutions to the inequality \( -{\rm div}_G \left( A(|\nabla_G u|) \nabla_G u \right) \geq u^q \quad {\rm in} \; \mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \) where \( A: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) satisfies some conditions below. Our results can be seen as a generalization of that in Mitidieri E & Pohozaev S.I. (2001). A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova, 234, 1–384, from the Laplace operator to the Grushin operator.