Liouville theorems for elliptic equations involving regional fractional Laplacian with order in (0, 1/2]

2020/07/11 by Chen, Huyuan, Wei, Yuanhong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.05775

Abstract

In this paper, some elliptic equation in a bounded open domain in ℝ^N (N≥ 2) with C² boundary ∂Ω is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the censored symmetric 2α-stable process in Ω. Probability theory asserts that the censored 2α-stable process can not approach the boundary when α∈(0,\frac12]. For α∈ (0,\frac12], our purpose in this article is to show that non-existence of solutions bounded from above or bounded from below for the particular Poisson problem (-Δ)^α_Ωu= 1 \rm in Ω and non-existence of nonnegative nontrivial solutions of the Lane-Emden equation (-Δ)^α_Ωu=u^p \rm in Ω, u=0 \rm on ∂Ω.

Related

• A Liouville theorem for non local elliptic equations
• Liouville type theorems for fractional elliptic problems
• Liouville theorem for the fractional Lane-Emden Equation in unbounded domain
• Refined regularity for nonlocal elliptic equations and applications
• The Dirichlet elliptic problem involving regional fractional Laplacian