Liouville theorems for $-\Delta_{m,g}u=\lambda e^u|\nabla u|^q$ in the full range of gradient exponents

Abstract

We study the equation \[ -\Delta_{m,g}u=\lambda e^u|\nabla u|^q \] on complete, connected, noncompact Riemannian manifolds of dimension $n\ge2$ with $\Ric_g\ge0$. For every $m>1$, $q\in\mathbb R$ and $\lambda>0$, 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 $m-1<q<m$, 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.