NONLINEAR LIOUVILLE-TYPE THEOREMS FOR GENERALIZED BAOUENDI-GRUSHIN OPERATOR ON RIEMANNIAN MANIFOLDS

We are concerned with diferential inequalities of the form (Formula Presented) where Mi (i = 1, 2) are complete noncompact Riemannian manifolds, (Formula Presented) is fixed, (Formula Presented) is the distance function on M1, dM2 (y0,.) is the distance function on M2, ΔMi is the Laplace-Beltrami operator on Mi, (Formula Presented) is a measurable function, and p > 1. Namely, we establish necessary conditions for existence of nontrivial weak solutions to the considered problem. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifolds Mi. Next, we discuss some special cases of potential functions V. The proof of our main result is based on the nonlinear capacity method and a result due to Bianchi and Setti (2018) about the construction of cut-off functions with controlled gradient and Laplacian, under certain assumptions on the Ricci curvatures of the manifolds. © 2023, Advances in Differential Equations. All Rights Reserved.