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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.

In this paper, we use the method of moving planes to derive the Harnack‐type inequality maxBRu·minB2Ru≤CRN−2α, for nonnegative solutions to fractional semilinear elliptic equations false(−normalΔfalse)αu=ffalse(ufalse),1em1emx∈B3Rfalse(0false),ufalse(xfalse)>0,1em1emx∈B3Rfalse(0false),ufalse(xfalse)=0,1em1emx∈double-struckRN∖B3Rfalse(0false), with 0

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