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Concave entire solutions of the Liouville equation
Abstract
We prove that every concave entire solution of the Liouville equation −Δu=e2u in Rn is one-dimensional, and we extend the result to a class of general semilinear equations under suitable conditions on the nonlinearity. In dimension two, we also provide a PDE-based proof of the known classification of quasiconcave solutions to planar Liouville solutions, without relying on complex-analytic methods developed in the previous work by Bergweiler-Eremenko-Langley \cite{BEL}
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BibTeX
@misc{gui2026concave,
title = {{Concave entire solutions of the Liouville equation}},
author = {Changfeng Gui and Qinfeng Li and Chunhua Wang and Lu Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33571},
eprint = {2610.33571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33571},
}
RIS
TY - UNPB
TI - Concave entire solutions of the Liouville equation
AU - Changfeng Gui
AU - Qinfeng Li
AU - Chunhua Wang
AU - Lu Xu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33571
UR - https://arxiv.org/abs/2610.33571
AB - We prove that every concave entire solution of the Liouville equation −Δu=e^2u in R^n is one-dimensional, and we extend the result to a class of general semilinear equations under suitable conditions on the nonlinearity. In dimension two, we also provide a PDE-based proof of the known classification of quasiconcave solutions to planar Liouville solutions, without relying on complex-analytic methods developed in the previous work by Bergweiler-Eremenko-Langley \cite{BEL}
ER -