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We are interested in the robustness of a Liouville type theorem for areaction diffusion equation in exterior domains. Indeed H. Berestycki, F. Hameland H. Matano (2009) proved such a result as soon as the domain satisfies somegeometric properties. We investigate here whether their result holds forperturbations of the domain. We prove that as soon as our perturbation is closeto the initial domain in the $C^{2,\alpha}$ topology the result remains truewhile it does not if the perturbation is not smooth enough.

Robustness for a Liouville type theorem in exterior domains