Liouville theorems for −Δm,gu=λeu|∇ u|q in the full range of gradient exponents
Abstract
We study the equation −Δm,gu=λeu|∇ u|q on complete, connected, noncompact Riemannian manifolds of dimension n≥2 with Ricg≥0. For every m>1, q∈ℝ and λ>0, we prove that every positive C1 weak solution is constant when q>0, while no such solution exists when q≤0. For q>m, we derive a local gradient estimate independent of the solution by Nash--Moser iteration, which leads to automatic boundedness and resolves the question raised by Lu and Zhu~\cite[Remark~1.6(B)]{LZ}. For m−1<q<m, we use an auxiliary function and a divergence inequality to extend the Liouville result beyond the range covered by existing gradient-inequality methods. The remaining cases are treated by suitable changes of variables and integral estimates. The conclusions also extend to real-valued solutions bounded below.
Related
- Gradient estimates and Liouville theorems for exponentially harmonic functions
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
- Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
- Hamilton's gradient estimates and Liouville theorems for ut=Δum+aulog u+bu on Riemannian manifolds
BibTeX
@misc{sun2026liouville,
title = {{Liouville theorems for $-Δ_m,gu=λe^u|\nabla u|^q$ in the full range of gradient exponents}},
author = {Yuhua Sun and Zhen Wang and Mengyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23029},
eprint = {2610.23029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23029},
}
RIS
TY - UNPB
TI - Liouville theorems for −Δ_m,gu=λe^u|∇ u|^q in the full range of gradient exponents
AU - Yuhua Sun
AU - Zhen Wang
AU - Mengyu Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23029
UR - https://arxiv.org/abs/2610.23029
AB - We study the equation −Δ_m,gu=λe^u|∇ u|^q on complete, connected, noncompact Riemannian manifolds of dimension n≥2 with Ric_g≥0. For every m>1, q∈ℝ and λ>0, we prove that every positive C^1 weak solution is constant when q>0, while no such solution exists when q≤0. For q>m, we derive a local gradient estimate independent of the solution by Nash--Moser iteration, which leads to automatic boundedness and resolves the question raised by Lu and Zhu~\cite[Remark~1.6(B)]{LZ}. For m−1<q<m, we use an auxiliary function and a divergence inequality to extend the Liouville result beyond the range covered by existing gradient-inequality methods. The remaining cases are treated by suitable changes of variables and integral estimates. The conclusions also extend to real-valued solutions bounded below.
ER -