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Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation

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Abstract

We study the multiplicity of concentrating solutions to the nonlinear fractional Kirchhoff equation

$$\begin{aligned} \left( \varepsilon ^{2s}a+\varepsilon ^{4s-3}b\int _{\mathbb R^3}|(-\Delta )^{\frac{s}{2}}u|^2dx\right) (-\Delta )^s u+V(x)u=f(u)~~\text{ in }~~\mathbb R^3, \end{aligned}$$

where \(\varepsilon >0\) is a positive parameter, \((-\Delta )^s\) is the fractional laplacian with \(s\in (\frac{3}{4},1), a,b\) are positive constants, and V is a positive potential such that \(\inf _{\partial \Lambda }V>\inf _{\Lambda }V\) for some open bounded subset \(\Lambda \subset \mathbb R^3.\) We relate the number of positive solutions with the topology of the set where V attains its minimum in \(\Lambda \). The proof is based on the Ljusternik–Schnirelmann theory.

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Acknowledgements

We would like to thank the anonymous referees for careful reading the manuscript and suggesting many valuable comments. X. He is supported by the National Natural Science Foundation of China (Grant No. 11771468, 11271386). W. Zou is supported by the National Natural Science Foundation of China (Grant No. 11771234, 11371212).

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Correspondence to Xiaoming He.

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X. He is supported by NSFC (11771468, 11271386) while W. Zou by NSFC (11771234, 11371212).

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He, X., Zou, W. Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation. manuscripta math. 158, 159–203 (2019). https://doi.org/10.1007/s00229-018-1017-0

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