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

Abstract This paper is concerned with the following problem involving almost critical growth \begin{equation*}\begin{cases}(-\Delta) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\u=0,& \text{on } \Sigma ,\\\end{cases}\end{equation*} where $2_{s}^{*}=\frac{2N}{N-2s}$ , $s\in(\frac{1}{2},1)$ , $N \gt 2s$ , Ω is a bounded domain in $\mathbb{R}^N$ , ɛ is a small parameter, and the boundary Σ is given in different ways according to the different definitions of the fractional Laplacian operator $(-\Delta)^{s}$ . The operator $(-\Delta)^{s}$ is defined in two types: the spectral fractional Laplacian and the restricted fractional Laplacian. For the spectral case, Σ stands for $\partial \Omega$ ; for the restricted case, Σ is $\mathbb{R}^{N}\setminus \Omega$ . 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.