Abstract Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation (−Δ)αu=upinΩ,u=0inRN∖Ω, $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ 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.

中文翻译：

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一般无界域中分数阶Lane-Emden方程的Liouville性质

摘要 本文的目的是考虑分数式 Lane-Emden 方程 (−Δ)αu=upinΩ,u=0inRN∖Ω, $$\begin{array}{} \displaystyle (-{\it\Delta })^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus { \it\Omega}, \end{array}$$ 其中 α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝN–1 × [0, +∞) 是一个满足Ωt的无界域： = {x′ ∈ ℝN–1 : (x′, t) ∈ Ω} t ≥ 0 具有递增的单调性，即 Ωt ⊂ Ωt′ 对于 t′ ≥ t。Ω∞ := limt→∞ Ωt 在ℝN-1 中的形状对于获得分数式 Lane-Emden 方程不存在正解起着重要作用。