NONEXISTENCE RESULT FOR QUASILINEAR ELLIPTIC INEQUALITIES INVOLVING THE GRUSHIN OPERATOR

DOI: 10.18173/2354-1059.2025-0001

Tóm tắt

In this paper, we establish a nonexistence result of positive solutions of the inequality -divG(jrGujp-2rGu) ≥ uq in RN = RN1 × RN2; where rG = (rx; jxjary) is the Grushin gradient, p; q > 1. Here (x; y) 2 RN1 × RN2 and a > 0. As a generalization of this result, we also establish a nonexistence result of positive solutions to the inequality -divGA(jrGuj)rGu ≥ uq in RN = RN1 ×RN2; where A : R+ ! R+ satisfies some conditions below. Our results can be seen as a generalization of that in [Mitidieri E & Pohozaev SI, (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.