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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.

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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