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On the Liouville property for divergence form operators. (English) Zbl 0912.31004
Let $$\sigma$$ be a strictly positive $$C^2$$ function on $${\mathbb{R}}^d$$ and let $$L=L[\sigma]=\nabla({\sigma}^2\nabla)$$ be the divergence form operator. The operator $$L$$ has the Liouville property if $$L\psi=0$$ and $$\sigma \psi$$ bounded imply that $$\psi$$ is constant. The main result of the paper under review is the following theorem showing that the Liouville property may fail when $$d\geq 3$$.
Theorem: Let $$d\geq 3$$. There exists a smooth strictly bounded function $$\sigma$$ on $${\mathbb{R}}^d$$ such that $$V=-{\sigma}^{-1}\Delta \sigma$$ is bounded, and the equation $$\nabla({\sigma}^2\nabla \phi)=0$$ has a bounded sign-changing solution.
The proof of the theorem is probabilistic and uses the diffusion generated by $$\frac 12 L[\sigma]$$. The constructed example also gives a negative answer to a problem raised in H. Berestycki, L. Caffarelli and L. Nirenberg [Further qualitative properties for elliptic equations in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 25, No. 1-2, 69-94 (1997)], concerning linear Schrödinger operators.

##### MSC:
 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 35J10 Schrödinger operator, Schrödinger equation
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