报告人：魏国栋（中山大学）

时间：2026年5月14日16：00-17：00

地点：腾讯会议 103 803 923

题目: Quasi-linear equation with integral bounded Ricci curvature and geometric applications

摘要: In this talk, I will discuss Liouville-type theorems and local gradient estimates for positive solutions to the quasilinear equation \Delta_p v+a v^q=0 on complete Riemannian manifolds under integral Ricci curvature bounds. We prove that the Liouville property remains valid when the negative part of the Ricci curvature is sufficiently small, in a suitable integral norm, relative to the Sobolev constant. In the special case \(p=2\), this yields an effective extension of the classical Gidas--Spruck Liouville theorem for the Lane--Emden equation.

 As a geometric application, we obtain a one-end criterion under a smallness condition on the integral norm of the negative part of the Ricci curvature. This result can be somehow regarded as an integral-curvature version of the gap theorem for ends of Cai, Colding, and Yang. We also establish local gradient estimates for positive solutions by means of a Nash--Moser iteration argument, thereby extending and improving a result of Petersen and Wei for harmonic functions.

 This talk is based on joint work with Professor Youde Wang and Liqin Zhang.