A counterexample to the Liouville property of some nonlocal problems

Abstract

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$$\int \_{\mathbb{R}}^N\setminus KJ(x-y)(u(y)-u(x))\mathrm{d}y+f(u(x))=0,x\in {\mathbb{R}}^N\setminus K,$$where $K\subset {\mathbb{R}}^N$ 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.