PreprintCC BY 4.0arXiv
Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities
Abstract
In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition −Δu = Ke2u & in 𝔹2, [2mm] (∂ u)/(∂ ν)+λ= keu & on ∂𝔹2, where K,k∈ℝ, λ∈(0,1] are constants, and ν denotes the outward unit normal on $ \partial\mathbb{B}^{2}$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of K and k within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev--Milin inequality, together with a corresponding deficit estimate.
Related
- Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk
- Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations
- Fundamental solutions of the fractional p-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity
- On a necessary condition for removing singularities of solutions of nonlinear elliptic inequalities
- Classification of radially symmetric solutions to a Hardy-Hénon equation with sublinear absorption and applications
BibTeX
@misc{cai2026classification,
title = {{Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities}},
author = {Xiaohan Cai and Abdolhakim Shouman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18762},
eprint = {2609.18762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18762},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities
AU - Xiaohan Cai
AU - Abdolhakim Shouman
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18762
UR - https://arxiv.org/abs/2609.18762
AB - In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition −Δu = Ke^2u & in 𝔹^2, [2mm] (∂ u)/(∂ ν)+λ= ke^u & on ∂𝔹^2, where K,k∈ℝ, λ∈(0,1] are constants, and ν denotes the outward unit normal on $ \partial\mathbb{B}^{2}$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of K and k within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev--Milin inequality, together with a corresponding deficit estimate.
ER -