Abstract This paper proves local gradient estimates on positive solutions to the following nonlinear elliptic equation Δ f u + a u ( log u ) α = 0 , where a and α are real constants, on complete weighted manifolds with Bakry–Emery Ricci tensor bounded from below. For applications, global estimates and some Liouville type theorems are derived.

中文翻译：

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加权非线性椭圆方程和刘维尔型定理的梯度估计

摘要 本文证明了以下非线性椭圆方程 Δ fu + au ( log u ) α = 0 的正解的局部梯度估计，其中 a 和 α 是实常数，在 Bakry-Emery Ricci 张量从下方有界的完整加权流形上。对于应用，推导出全局估计和一些刘维尔类型定理。