Global and local properties of solutions of elliptic equations with a nonlinear term involving the product of the function and its gradient

Abstract

We study the global and local properties of positive solutions to the quasi-linear elliptic equation: \d u+|\nabla u|^q u^p=0,\quad x\in \O\subset \mathbb{R}^n,\nonumber where $q\ge 0$ and $p\in\mathbb{R}$. Our contributions are twofold: 1. Based on an optimal and new identity for the modulus squared of the logarithmic gradient, we establish optimal and improved Liouville theorems for global positive solutions, and generalize these findings to the framework of Riemannian manifolds. 2. Based on a newly discovered mutual control relationship of two nonlinear iterms, for all index pairs \( (p, q) \) where the Liouville theorem holds, we derive several optimal gradient estimates for local positive solutions. As a direct corollary, we obtain the corresponding Harnack inequality. These results strengthen the related conclusions in Bidaut-V\'eron--Garc\'ia-Huidobro--V\'eron \cite {BGV} from both global and local perspectives.