Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
Year: 2026
Type: preprint
Abstract: In this paper, we study local and global properties of positive solutions to the equation $-Δu=u^p|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.
Source: arXiv (Cornell University)
Authors: Liang, Wenguo, Zhang, Zhengce
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2873516585, G9000363291
OpenAlex ID: W7213412919