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Liouville theorems for quasilinear differential inequalities involving gradient nonlinearity term on manifolds

2021年4月30日13:30-14:30  腾讯会议ID：491 801 880

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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 a complete classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville 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 values, even in the Euclidean space.

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