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# On the Liouville property for divergence form operators

Published:1998-06-01
Printed: Jun 1998
• Martin T. Barlow
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## Abstract

In this paper we construct a bounded strictly positive function $\sigma$ such that the Liouville property fails for the divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in addition $\Delta \sigma/\sigma$ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schr\"odinger operators.
 MSC Classifications: 31C05 - Harmonic, subharmonic, superharmonic functions 60H10 - Stochastic ordinary differential equations [See also 34F05] 35J10 - Schrodinger operator [See also 35Pxx]