Science China Mathematics

, Volume 62, Issue 5, pp 853–890 | Cite as

Ground state solutions for a class of fractional Kirchhoff equations with critical growth

  • Xiaoming HeEmail author
  • Wenming Zou


In this paper, we study the effect of lower order perturbations in the existence of positive solutions to the fractional Kirchhoff equation with critical growth \(\left(a+b\int_{\mathbb{R}^{3}}|(-\triangle)^\frac{s}{2}u|^{2}dx\right)(-\triangle)^{s}u+V(x)u=\mu|\mu^{p-1}u+|u|^{2_s^*}-^{2}u, x\epsilon{\mathbb{R^3}},\) where a; b > 0 are constants, μ > 0 is a parameter, \(s\in(\frac{3}{4},1),\;1<p<2_s^*-1=\frac{3+2s}{3-2s}\), and V : ℝ3 → ℝ is a continuous potential function. For suitable assumptions on V, we show the existence of a positive ground state solution, by using the methods of the Pohozaev-Nehari manifold, Jeanjean’s monotonicity trick and the concentration-compactness principle due to Lions (1984).


fractional Kirchhoff equations ground state solutions Pohozaev-Nehari manifold critical Sobolev exponent 


35J60 35J70 


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The first author was supported by National Natural Science Foundation of China (Grant Nos. 11771468 and 11271386). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11771234 and 11371212). The authors thank the referees for their careful reading of the manuscript and helpful comments.


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of ScienceMinzu University of ChinaBeijingChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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