光滑度量测度空间上方程△_p,fu + au^r|?u|^s=0解的线性化算子处理

Treatment of the Linearized Operator for Solutions to the Equation △_p,fu + au^r|?u|^s=0 on Smooth Metric Measure Spaces

摘要: 在这篇论文中，我们考虑光滑度量测度空间上的加权p拉普拉斯方程△_p,fu + au^r|∇u|^s=0，其中p > 1,a,r和s为常数，且加权p拉普拉斯算子定义为△_p,fu= e^f div (e^-f|∇u|^p-²∇u)。随后，我们利用不等式处理此方程解的线性化算子。在关于a,p,r和 s的一些假设条件下，我们导出了此方程可用于做梯度估计和刘维尔型定理的线性化算子。记∇,△和Hess 分别为梯度、拉普拉斯和海森算子，dυ为黎曼体积测度。

Abstract: In this paper, we consider the weighted p-Laplace equation △_p,fu + au^r|∇u|^s=0 on smooth metric measure spaces, wherep > 1, and a, r, s are constants, with the weighted p-Laplace operator defined as △_p,fu= e^f div (e^-f|∇u|^p-²∇u). Subsequently, we analyze the linearized operator for solutions to this equation using inequality techniques. Under certain assumptions on a, p, r, and s, we derive a linearized operator for this equation that can be applied to gradient estimates and Liouville-type theorems. Let ∇, △, and Hess denote the gradient, Laplacian, and Hessian operators, respectively, and let dv represent the Riemannian volume measure.