Robustness for a Liouville Type Theorem in Exterior Domains

We are interested in the robustness of a Liouville type theorem for a reaction diffusion equation in exterior domains. Indeed Berestycki et al. (Commun. Pure Appl. Math., 62(6):729–788, 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 C2,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} topology the result remains true while it does not if the perturbation is not smooth enough.