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Non-degeneracy and local uniqueness of bubbling solutions for fractional almost critical problems
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 03 November 2025, pp. 1-45
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- Article
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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.
2 - Perturbation Methods
-
- Book:
- Singularly Perturbed Methods for Nonlinear Elliptic Problems
- Published online:
- 28 January 2021
- Print publication:
- 18 February 2021, pp 54-128
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- Chapter
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Singularly Perturbed Methods for Nonlinear Elliptic Problems
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- Published online:
- 28 January 2021
- Print publication:
- 18 February 2021
