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arXiv · 2610.33571

Concave entire solutions of the Liouville equation

Changfeng Gui·Qinfeng Li·Chunhua Wang·Lu Xu
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Abstract

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}

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Changfeng Gui, Qinfeng Li, Chunhua Wang, Lu Xu. 2026-10-24. Concave entire solutions of the Liouville equation. https://arxiv.org/abs/2610.33571

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