PreprintCC BY 4.0arXiv
Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
Abstract
We study local and global properties of positive solutions to the equation −Δu=up+M|∇ u|q in a domain Ω of ℝN, where p,q are parameters and M>0. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate auxiliary functions over various regions and employing the maximum principle, we derive the local gradient estimates for all (p,q)∈ ℝ2, and further establish Liouville-type theorems. As an application, we acquire universal estimates for local solutions of elliptic equations with general nonlinearities. Our results extend partial conclusions established in Bidaut-Véron, Garcia-Huidobro and Véron [Math. Ann. 378 (1-2) (2020) 13-56].
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BibTeX
@misc{liang2026gradient,
title = {{Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source}},
author = {Wenguo Liang and Zhengce Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13738},
eprint = {2609.13738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13738},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
AU - Wenguo Liang
AU - Zhengce Zhang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13738
UR - https://arxiv.org/abs/2609.13738
AB - We study local and global properties of positive solutions to the equation −Δu=u^p+M|∇ u|^q in a domain Ω of ℝ^N, where p,q are parameters and M>0. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate auxiliary functions over various regions and employing the maximum principle, we derive the local gradient estimates for all (p,q)∈ ℝ^2, and further establish Liouville-type theorems. As an application, we acquire universal estimates for local solutions of elliptic equations with general nonlinearities. Our results extend partial conclusions established in Bidaut-Véron, Garcia-Huidobro and Véron [Math. Ann. 378 (1-2) (2020) 13-56].
ER -