Liouville theorems for $-Δ_{m,g}u=λe^u|\nabla u|^q$ in the full range of gradient exponents
Year: 2026
Type: preprint
Abstract: We study the equation \[ -Δ_{m,g}u=λe^u|\nabla u|^q \] on complete, connected, noncompact Riemannian manifolds of dimension $n\ge2$ with $\Ric_g\ge0$. For every $m>1$, $q\in\mathbb R$ and $λ>0$, we prove that every positive $C^1$ weak solution is constant when $q>0$, while no such solution exists when $q\le0$. 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
Source: arXiv (Cornell University)
Authors: Sun, Yuhua, Wang, Zhen, Zhang, Mengyu
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222916645