Gradient estimates and a Liouville type theorem for the Schrödinger operator
Let \(M\) be a complete Riemannian manifold without boundary. Assume that the Ricci curvature of \(M\) is nonnegative. Let \(h\) be a \(C^2\) function defined on \(M\) such that (i) \(\Delta h \geq 0\), (ii) \(h > 0\) at some point, (iii) \(\lim_{r \to \infty} r^{-2} \inf_{x \in B_p(r)} h(x) = 0\), where \(B_p(r)\) is the geodesic ball of radius \(r\) around some fixed point \(p \in M\). Under these assumptions, the author proves that the equation \(\Delta u + hu = 0\) has no positive smooth solution on \(M\). This generalizes earlier results of P. Li and S.-T. Yau, and of J. Li.
scientific article; zbMATH DE number 763986