Liouville property of fractional Lane-Emden equation in general unbounded domain

Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝN–1 × [0, +∞) is an unbounded domain satisfying that Ωt := {x′ ∈ ℝN–1 : (x′, t) ∈ Ω} with t ≥ 0 has increasing monotonicity, that is, Ωt ⊂ Ωt′ for t′ ≥ t. The shape of Ω∞ := limt→∞ Ωt in ℝN–1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.