Liouville type results for a nonlocal obstacle problem

Abstract

This paper is concerned with qualitative properties of solutions to nonlocal reaction–diffusion equations of the form 0true∫RN∖KJ(x−y)0.16em(ufalse(yfalse)−ufalse(xfalse))0.16emnormaldy+f(ufalse(xfalse))=0,1emx∈RN∖K, set in a perforated open set double-struckRN∖K, where K⊂double-struckRN 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.

A Note on Liouville type results for a fractional obstacle problem

“…Similar Liouville type property was recently obtained for continuous solution of (1.9) when the singular kernel 1 |z| n+2s is replaced by a non negative integrable kernel J, i.e. J ∈ L 1 (R), see [4]. More precisely, if J is assume to satisfy the assumptions below (2.13) J ∈ L 1 (R n ) is a non-negative, radially symmetric kernel with unit mass, there are 0 r 1 < r 2 such that J(x) > 0 for a.e.…”

“…The next lemma is concerned with a strong maximum principle. These comparison principles are in essence identical to the one derived in [4] and as such we point the interested reader to [4] for a detailed proof of these results.…”

“…Again the proof of this claim is an elementary adaptation of a proof done in [4] that for the sake of clarity we give the details.…”

“…The proof of this Lemma being an elementary adaptation of the argument used in [4], we will refer to [4] for its proof.…”

“…Let us fix s ∈ 1 2 , 1 and let K, f , and u be as in Theorem 2.3. Let us now follow the argument developed in [4]. Firstly, without loss of generality, one can assume by (1.11) that f is extended to a C 1 (R) function satisfying (3.19).…”