On critical Kirchhoff problems driven by the fractional Laplacian

Appolloni, Luigi; Bisci, Giovanni Molica; Secchi, Simone

doi:10.1007/s00526-021-02065-8

Cited by 7 publications

(1 citation statement)

References 30 publications

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“…Recently, problem similar to (1.1) has been extensively investigated by many authors using different techniques and producing several relevant results (see, e.g. [1,2,3,15,30,31,32,41]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Yang¹

2022

Preprint

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This paper is twofold. In the first part, combining the nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of multi-peak positive solutions to the singularly perturbation problems − 1 and some mild assumptions on the function V . The main difficulties are from the nonlocal operator mixed the nonlocal term, which cause the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single fractional Kirchhoff equation. In the second part, under some assumptions on V , we show the local uniqueness of positive multi-peak solutions by using the local Pohozǎev identity.

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“…Recently, problem similar to (1.1) has been extensively investigated by many authors using different techniques and producing several relevant results (see, e.g. [1,2,3,15,30,31,32,41]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Yang¹

2022

Preprint

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Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

Yang

2022

J Geom Anal

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In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem $$\begin{aligned} \Big (a+b{\int _{\mathbb {R}^{N}}}|(-\Delta )^{\frac{s}{2}}u|^2\mathrm{{d}}x\Big )(-\Delta )^su+u=u^p,\quad \text {in}\ \mathbb {R}^{N}, \end{aligned}$$ ( a + b ∫ R N | ( - Δ ) s 2 u | 2 d x ) ( - Δ ) s u + u = u p , in R N , where $$a,b>0$$ a , b > 0 , $$0<s<1$$ 0 < s < 1 , $$1<p<\frac{N+2s}{N-2s}$$ 1 < p < N + 2 s N - 2 s and $$(-\Delta )^s$$ ( - Δ ) s is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions $$N>4s$$ N > 4 s , i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low dimensional fractional Kirchhoff equation. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems.

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Fractional Kirchhoff-Type and Method of Sub-supersolutions

Sousa

2023

Differ Equ Dyn Syst

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On critical Kirchhoff problems driven by the fractional Laplacian

Cited by 7 publications

References 30 publications

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Non-degeneracy of Positive Solutions for Fractional Kirchhoff Problems: High Dimensional Cases

Fractional Kirchhoff-Type and Method of Sub-supersolutions

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