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

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.