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

Advances in Nonlinear Analysis

article

Ying Wang¹  Yuanhong Wei² 

[1] Department of Mathematics, Jiangxi Normal University;School of Mathematics, Jilin University

关键词: Fractional Laplacian;    Lane-Emden equation;    Nonexistence;   

DOI  :  10.1515/anona-2020-0147

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation (− Δ )α u=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 ⊂ ℝ N –1 × [0, +∞) is an unbounded domain satisfying that Ω t := { x ′ ∈ ℝ N –1 : ( x ′, t ) ∈ Ω } with t ≥ 0 has increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The shape of Ω ∞ := lim t →∞ Ω t in ℝ N –1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.

【 授权许可】

CC BY