An Omori-Yau maximum principle for semi-elliptic operators and Liouville-type theorems

We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold $M$ to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued $C^2$ function $f$ on $M$ satisfying $Lf\ge F(f)+H(\mathrm{\mid }\mathrm{\nabla }f\mathrm{\mid })$ for real-valued continuous functions $F$ and $H$ on $\mathbb{R}$ such that $H(0)=0$.