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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Liouville type theorem
for the Schrödinger operator


Author: Antonios D. Melas
Journal: Proc. Amer. Math. Soc. 127 (1999), 3353-3359
MSC (1991): Primary 58G03; Secondary 35J10, 58G11
Published electronically: June 17, 1999
MathSciNet review: 1623036
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Abstract: In this paper we prove that the equation $\Delta u(x)+h(x)u(x)=0$ on a complete Riemannian manifold of dimension $n$ without boundary and with nonnegative Ricci curvature admits no positive solution provided that $h$ is a $C^2$ function satisfying $\limsup _{r\rightarrow\infty}r^{-2}\inf _{x\in B_p(r)}h(x) \geq-b_na^2$ and $\Delta h(x)\geq-c_na^2$ where $0\leq a<\sup _{x\in M}h(x)$, and $b_n,c_n$ are constants depending only on the dimension, thus generalizing similar results in P. Li and S. T. Yau (Acta Math. 156 (1986), 153-201), J. Li (J. Funct. Anal. 100 (1991), 233-256) and E. R. Negrin (J. Funct. Anal. 127 (1995), 198-203) in all of which $h$ is assumed to be subharmonic. We also give a generalization in case the Ricci curvature of $M$ is not necessarily positive but its negative part has quadratic decay under the additional assumption that $h$ is unbounded from above.


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Additional Information

Antonios D. Melas
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis, Athens 157-84, Greece
Email: amelas@math.uoa.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05026-1
PII: S 0002-9939(99)05026-1
Received by editor(s): January 12, 1998
Published electronically: June 17, 1999
Communicated by: Peter Li
Article copyright: © Copyright 1999 American Mathematical Society