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

Yuanhong Wei,Ying Wang

doi:10.1515/anona-2020-0147

Abstract

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

Highlights

In this paper, we consider Liouville property for the fractional Lane-Emden equation Ω, (1.1)where α ∈ (, ), p >, Ω is an unbounded domain in RN with N ≥, and (−∆)α with α ∈ (, ) is the fractional Laplacian de ned in the principle value sense, (−∆)α u(x) = cN,α lim ε→ +u(x) − u(x + |z|N+ α z) d z, RN \Bε( )here Bε( ) is the ball with radius ε centered at the origin and cN,α > is the normalized constant
Where α ∈ (, ), p >, Ω is an unbounded domain in RN with N ≥, and (−∆)α with α ∈ (, ) is the fractional Laplacian de ned in the principle value sense, (−∆)α u(x) cN,α lim ε→ +
The Liouville theorem for Lane-Emden equation has attracted a lot of attentions by many mathematician by the application in the derivation of uniform bound via blowing-up analysis

Summary

Introduction

We consider Liouville property for the fractional Lane-Emden equation. As far as we know, the nonexistence of elliptic equation depends on the shape of limit domain at in nity in some direction. In [15] it develops the method of moving plane in fractional setting to obtain the classi cation of critical elliptic equations in an integral form and it is used to obtain nonexistence of bounded solutions for semilinear elliptic equations in half space [12, 13, 16] subject to various boundary type conditions. It is worth noting that star-shaped domain with respect to in nity is involved in obtaining the nonexistence of fractional Lane-Emden equation in [14] and it is an important notation in our derivation of nonexistence to (1.1).

The proof of nonexistence results

Using and proceeding as in