涉及函数与其梯度乘积的半线性椭圆方程的Liouville型定理与普适估计
Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过引入线性算子构造微分不等式，研究半线性椭圆方程正解的梯度估计与Liouville型定理，并将参数条件推广至更广范围，获得局部解的普适估计。
AI中文摘要：
本文研究定义在$\mathbb R^N$的区域$\Omega$中方程$-\Delta u=u^p|\nabla u|^q$的正解的局部与全局性质，其中$p$和$q$为参数。我们引入一个线性算子来构造微分不等式，从而获得梯度估计，并进一步建立Liouville型定理。作为应用，我们推导出局部解的普适估计。我们的一些结果是新的，因为我们把He、Hu和Wang [Math. Z. 313 (2026), No. 6]所考虑的$p+q<(N+3)/(N-1)$条件推广到了更广泛的参数范围。
英文摘要：
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.
