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Science China Mathematics

, Volume 60, Issue 9, pp 1647–1660 | Cite as

Existence of solutions for a critical fractional Kirchhoff type problem in ℝ N

  • MingQi Xiang
  • BinLin Zhang
  • Hong Qiu
Articles

Abstract

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity:
$${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$
where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

Keywords

fractional Laplacian Kirchhoff problem mountain pass theorem Ekeland variational principle 

MSC(2010)

35R11 35A15 47G20 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601515 and 11401574), the Fundamental Research Funds for the Central Universities (Grant No. 3122015L014), and the Doctoral Research Foundation of Heilongjiang Institute of Technology (Grant No. 2013BJ15). The authors thank Professor Giovanni Molica Bisci for useful suggestions and comments.

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of ScienceCivil Aviation University of ChinaTianjinChina
  2. 2.Department of MathematicsHeilongjiang Institute of TechnologyHarbinChina

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