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

Yuhua Sun; Fanheng Xu

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arXiv · 2102.02073

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

Yuhua Sun·Fanheng Xu

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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.

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Yuhua Sun, Fanheng Xu. 2021-02-03. Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds. https://arxiv.org/abs/2102.02073

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