Gradient Estimates and a Liouville Type Theorem for the Schrِdinger Operator

In this paper, we derive a Liouville type theorem on a complete Riemannian manifold without boundary and with nonnegative Ricci curvature for the equation Δu(x)+h(x)u(x)=0, where the conditions limr→∞r−1.supx ∈ Bp(r)|Δh(x)| = 0 and h ≥ 0 imposed by P. Li and S. -T. Yau (Acta Math. 156 (1986), 153-201) and Jiayu Li (J. Funct. Anal. 100 (1991), 233-256), respectively, are replaced by a weaker condition than both of them, namely, limr→∞r−2·infx∈Bp(r)h(x) = 0.