Non-degeneracy and local uniqueness of bubbling solutions for fractional almost critical problems

This paper is concerned with the following problem involving almost critical growth {equation*}{cases}(-Δ) ^su=|u|^{2ₛ*-2-ε}u,& in ,\=0,& on Σ ,\\{cases}{equation*} where 2ₛ*=2N/N-2s , s∈(1/2,1) , N > 2s , Ω is a bounded domain in RN , ɛ is a small parameter, and the boundary Σ is given in different ways according to the different definitions of the fractional Laplacian operator (-Δ)ˢ . The operator (-Δ)ˢ is defined in two types: the spectral fractional Laplacian and the restricted fractional Laplacian. For the spectral case, Σ stands for ∂ Ω ; for the restricted case, Σ is RN Ω . Firstly, we provide a positive confirmation of the fractional Brezis–Peletier conjecture, that is, the above almost critical problem has a single bubbling solution concentrating around the non-degenerate critical point of the Robin function. Furthermore, the non-degeneracy andlocal uniqueness of this bubbling solution are established.