Liouville theorems for stable solutions to the mean curvature equation
Abstract
We study stable solutions of the mean curvature equation −div((∇ u)/(√(1+|∇ u|2))) =|u|p−1u in domains of ℝn. We establish Liouville-type results in three settings. First, for globally stable solutions in ℝn, we prove that u≡0 if 1<p<pJL, where pJL is the Joseph--Lundgren exponent. Second, for solutions in ℝn that are stable outside a compact set, we show that if n>2 and p=pS, where pS is the Sobolev critical exponent, then u≡0. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if n>2, 1<p≤ pS, and the solution is stable outside a compact set, then u≡0. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.
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BibTeX
@misc{xu2026liouville,
title = {{Liouville theorems for stable solutions to the mean curvature equation}},
author = {Fanheng Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14652},
eprint = {2610.14652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14652},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Liouville theorems for stable solutions to the mean curvature equation
AU - Fanheng Xu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14652
UR - https://arxiv.org/abs/2610.14652
AB - We study stable solutions of the mean curvature equation −div((∇ u)/(√(1+|∇ u|^2))) =|u|^p−1u in domains of ℝ^n. We establish Liouville-type results in three settings. First, for globally stable solutions in ℝ^n, we prove that u≡0 if 1<p<p_JL, where p_JL is the Joseph--Lundgren exponent. Second, for solutions in ℝ^n that are stable outside a compact set, we show that if n>2 and p=p_S, where p_S is the Sobolev critical exponent, then u≡0. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if n>2, 1<p≤ p_S, and the solution is stable outside a compact set, then u≡0. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.
ER -