Acta mathematica scientia,Series B ›› 2018, Vol. 38 ›› Issue (6): 1712-1730.

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SIGN-CHANGING SOLUTIONS FOR THE STATIONARY KIRCHHOFF PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN IN RN

Kun CHENG1, Qi GAO2   

  1. 1 Department of Information Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333403, China;
    2 Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China
  • Received:2017-08-09 Revised:2018-01-11 Online:2018-12-25 Published:2018-12-28
  • Contact: Qi GAO E-mail:gaoq@whut.edu.cn
  • Supported by:
    The second author was supported by the NSFC (11501231), and the "Fundamental Research Funds for the Central Universities" (WUT2017IVA077, 2018IB014).

Abstract: In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution ub for the given problem. Moreover, we show that the energy of ub is strictly larger than twice the ground state energy. We also give a convergence property of ub as b↘ 0, where b is regarded as a positive parameter.

Key words: Kirchhoff equation, fractional Laplacian, sign-changing solutions

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