Liouville type results for a nonlocal obstacle problem

This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.