Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1612-1621.

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Existence of Multiple Solutions for a Class of Fractional Schrödinger-Kirchhoff Equation

Jianli Li1,Anran Li2,*(),Chongqing Wei2,Gang Li3   

  1. 1 Department of Applied Mathematics, Taiyuan Institute, Taiyuan 030032
    2 School of Mathematical Sciences, Shanxi University, Taiyuan 030006
    3 School of Mathematical Sciences, Yangzhou University, Jiangsu Yangzhou 225002
  • Received:2019-12-03 Online:2020-12-26 Published:2020-12-29
  • Contact: Anran Li
  • Supported by:
    the NSFC(11701346);the NSFC(11871064)


In this article, we use variational method and the critical point theory to study the existence of multiple solutions for a class of Schrödinger-Kirchhoff equation involving the fractional $p$-Laplacian operator where $ \lambda\in \ {\Bbb R} , 0<s<1<r<p<2, ps<N, (-\Delta)_p^{s} $ is the fractional p-Laplacian operator. Under certain assumptions, we first show the existence of multiple high energy solutions by means of symmetric mountain pass theorem. Secondly, by using dual fountain theorem, we prove that the above equation has a sequence of negative energy solution, whose energy converges to 0.

Key words: Schrödinger-Kirchhoff equation, Fractional p-Laplacian operator, Symmetric mountain pass theorem, Dual fountain theorem

CLC Number: 

  • O176.3