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

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Abstract

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).

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Acknowledgements

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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  1. College of Science, Minzu University of China, Beijing, 100081, China

    Xiaoming He

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wenming Zou

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  1. Xiaoming He
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  2. Wenming Zou
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Correspondence to Xiaoming He.

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He, X., Zou, W. Ground state solutions for a class of fractional Kirchhoff equations with critical growth. Sci. China Math. 62, 853–890 (2019). https://doi.org/10.1007/s11425-017-9399-6

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