A counterexample to the Liouville property of some nonlocal problems

Julien Brasseur; Jérôme Coville

doi:10.1016/j.anihpc.2019.12.003

JournalsaihpcVol. 37, No. 3pp. 549–579

A counterexample to the Liouville property of some nonlocal problems

Julien Brasseur
EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
Jérôme Coville
BioSP, INRAE, 84914, Avignon, France

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Abstract

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form

R^{N} ∖ K \int J (x - y) (u (y) - u (x)) d y + f (u (x)) = 0, x \in R^{N} ∖ K,

where $K \subset R^{N}$ is a bounded compact set, called an “obstacle”, and f is a bistable nonlinearity. When K is convex, it is known that solutions ranging in $[0, 1]$ and satisfying $u (x) \to 1$ as $∣ x ∣ \to \infty$ must be identically 1 in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data f and J for which this property fails.

Cite this article

Julien Brasseur, Jérôme Coville, A counterexample to the Liouville property of some nonlocal problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 549–579

DOI 10.1016/J.ANIHPC.2019.12.003