A counterexample to the Liouville property of some nonlocal problems
Brasseur, Julien1 ; Coville, Jérôme2
1 EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
2 BioSP, INRAE, 84914, Avignon, France
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 549-579.
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In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form

∫RN∖KJ(x−y)(u(y)−u(x))dy+f(u(x))=0,x∈RN∖K,
where K⊂RN 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)→1 as |x|→∞ 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.

DOI : 10.1016/j.anihpc.2019.12.003
Mots-clés : Nonlocal diffusion, Liouville property, Bistable nonlinearity, Counterexample, Positive stationary solution, Calculus of variations
Brasseur, Julien 1 ; Coville, Jérôme 2

1 EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
2 BioSP, INRAE, 84914, Avignon, France 
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     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Brasseur, Julien; Coville, Jérôme. A counterexample to the Liouville property of some nonlocal problems. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 549-579. doi : 10.1016/j.anihpc.2019.12.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.12.003/

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