数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1612-1621.

• 论文 • 上一篇    下一篇

一类分数阶Schrödinger-Kirchhoff方程多重解的存在性

李建利1,李安然2,*(),魏重庆2,李刚3   

  1. 1 太原学院应用数学系 太原 030032
    2 山西大学数学科学学院 太原 030006
    3 扬州大学数学科学学院 江苏扬州 225002
  • 收稿日期:2019-12-03 出版日期:2020-12-26 发布日期:2020-12-29
  • 通讯作者: 李安然 E-mail:lianran@sxu.edu.cn
  • 基金资助:
    国家自然科学基金(11701346);国家自然科学基金(11871064)

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 E-mail:lianran@sxu.edu.cn
  • Supported by:
    the NSFC(11701346);the NSFC(11871064)

摘要:

利用变分方法和临界点理论讨论了一类带有分数阶$p$-拉普拉斯算子的Schrödinger-Kirchhoff方程多重解的存在性 其中$ \lambda\in \ {\Bbb R} , 0<s<1<r<p<2, ps<N, (-\Delta)_p^{s} $表示分数阶p -拉普拉斯算子.首先, 利用对称山路定理得到该方程无穷多高能量解的存在性.其次, 利用对偶喷泉定理证明了上述方程有一列趋于0的负能量解.

关键词: Schrödinger-Kirchhoff方程, 分数阶p-拉普拉斯算子, 对称山路定理, 对偶喷泉定理

Abstract:

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

中图分类号: 

  • O176.3