Gradient estimates and a Liouville type theorem for the Schrödinger operator
type: Item
instance of: scholarly article
title: Gradient estimates and a Liouville type theorem for the Schrödinger operator
zbMATH Open document ID: 0842.58078
published in: Journal of Functional Analysis
publication date: 1996-08-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J99
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 763986
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Riemannian manifold
author: E. R. Negrin
reviewed by: Paul Godin
MaRDI profile type: Publication
full work available at URL: jfan.1995.1008
OpenAlex ID: W1996024624
DOI: 10.1006/JFAN.1995.1008
description: scientific article; zbMATH DE number 763986