Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

TL;DR

This paper establishes Liouville-type theorems for quasilinear differential inequalities involving gradient terms on manifolds, identifying sharp volume growth conditions for existence and nonexistence of positive solutions, including new results for negative parameter pairs.

Contribution

It provides new Liouville theorems for inequalities with gradient nonlinearities on manifolds, covering cases with negative parameters, and determines sharp volume growth conditions.

Findings

Different volume growth conditions lead to existence or nonexistence of solutions.

Results are sharp in most cases, indicating optimal conditions.

New insights for inequalities with negative parameter pairs, even in Euclidean space.

Abstract

We investigate the nonexistence and existence of nontrivial positive solutions to $\Delta_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for $(p, q)$ of negative pair, even in the Euclidean space.