Effect of the Domain Geometry on the Solutions to Fractional Brezis-Nirenberg Problem

Qiaoyu Tian,Yonglin Xu

doi:10.1155/2019/1093804

Abstract

In this paper, we consider the Brezis-Nirenberg problem for the nonlocal fractional elliptic equation Aαux=NN-2αuxp+εux, x∈Ω, ux>0, x∈Ω, u(x)=0, x∈∂Ω, where 0<α<1 is fixed, p=N+2α/N-2α, ε is a small parameter, and Ω is a bounded smooth domain of RN(N≥4α). Aα denotes the fractional Laplace operator defined through the spectral decomposition. Under some geometry hypothesis on the domain Ω, we show that all solutions to this problem are least energy solutions.

Highlights

Introduction and Main ResultsIn the famous paper of Brezis and Nirenberg [1], they studied the following nonlinear critical elliptic partial differential equation:−Δu (x) = N (N − 2) u (x)(N+2)/(N−2) + εu (x), u (x) > 0, x ∈ Ω, x ∈ Ω, (1)u (x) = 0, x ∈ ∂Ω, where Ω is a bounded domain in RN(N ≥ 3)
Rey [3] and Han [4] established the asymptotic behavior of positive solutions to problem (1) by different methods independently
We study the following nonlocal BrezisNirenberg problem: Aαu (x) = N (N − 2α) u (x)p + εu (x), x ∈ Ω, u (x) > 0, x ∈ Ω, (2)

Summary

Introduction and Main Results

In the famous paper of Brezis and Nirenberg [1], they studied the following nonlinear critical elliptic partial differential equation:. U (x) = 0, x ∈ ∂Ω, where Ω is a bounded domain in RN(N ≥ 3) They proved that problem (1) has a positive nontrivial solution provided N ≥ 4 and ε ∈ (0, λ1), where λ1 is the first eigenvalue of −Δ in Ω. This result was extended by Capozzi et al [2] for every parameter ε. Based on fractional harmonic extension formula of Caffarelli and Silvestre [14] (see Cabreand Tan [15] ), Choi et al [9] studied the asymptotic behavior of least energy solutions u to problem (2); that is, u satisfies lim ε󳨀→0.

Useful Definitions

Proof of Theorem 1