Zhen Wang

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

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...