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PreprintCC BY 4.0arXiv

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=up|∇ u|q in a domain Ω of ℝN, 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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Cite

BibTeXRIS
BibTeX
@misc{liang2026liouvilletype,
  title = {{Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient}},
  author = {Wenguo Liang and Zhengce Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17149},
  eprint = {2609.17149},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17149},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
AU  - Wenguo Liang
AU  - Zhengce Zhang
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17149
UR  - https://arxiv.org/abs/2609.17149
AB  - In this paper, we study local and global properties of positive solutions to the equation −Δu=u^p|∇ u|^q in a domain Ω of 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.
ER  -