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].