Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian (−Δ)Ωsu+Vu=h1up+h2in Ω,u=0on ∂Ω, where s∈(0,1), p>0, h1,h2 are nonnegative functions and Ω⊂RN−1×[0,+∞) with N≥2, is an unbounded domain satisfying Ωt:={x′∈RN−1:(x′,t)∈Ω} with t≥0 having an increasing monotonicity, that is, Ωt⊂Ωt′ for t′≥t. The potential V(x′,t) decays as t→+∞. The properties of the limit domain Ω∞:=limt→∞Ωt in RN−1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s∈(0,12], we provide a surprising nonexistence result if Ω∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s∈(0,12].