A Note on Liouville type results for a fractional obstacle problem

Abstract

This note is a synthesis of my thoughts on some questions that have emerged during the MATRIX event “Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type” concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form

$$ \fancyscript{L}[u](x) + f(u(x)) \, = 0,\,\,\,\,{\text{for }}x \in \overline{{{\mathbb{R}}^{n} \backslash K,}} $$

where $K \subset {\mathbb{R}}^{N}$ is a bounded smooth compact “obstacle”, $\fancyscript{L}$ is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator $\fancyscript{L}$ is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity have been recently established by Brasseur, Coville, Hamel and Valdinoci [4]. Here, we show that for a bounded smooth convex obstacle K, similar Liouville type results hold true when the operator $\fancyscript{L}$ is the regional s-fractional Laplacian.