  We are interested in the robustness of a Liouville type theorem for a
reaction diffusion equation in exterior domains. Indeed H. Berestycki, F. Hamel
and H. Matano (2009) proved such a result as soon as the domain satisfies some
geometric properties. We investigate here whether their result holds for
perturbations of the domain. We prove that as soon as our perturbation is close
to the initial domain in the $C^{2,\alpha}$ topology the result remains true
while it does not if the perturbation is not smooth enough.
