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arXiv · 2609.17149

Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient

Wenguo Liang·Zhengce Zhang
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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.

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Wenguo Liang, Zhengce Zhang. 2026-09-15. Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient. https://arxiv.org/abs/2609.17149

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