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 C2C2 function f on M   satisfying Lf⩾F(f)+H(|∇f|)Lf⩾F(f)+H(|∇f|) for real-valued continuous functions F and H   on RR such that H(0)=0H(0)=0.