Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (3): 1131-1160.doi: 10.1007/s10473-023-0309-y

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THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS*

Leilei CUI, Jiaxing GUO, Gongbao LI   

  1. Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • Received:2021-09-24 Online:2023-06-25 Published:2023-06-06
  • Contact: Gongbao LI, E-mail: ligb@mail.ccnu.edu.cn
  • About author:Leilei CUI, E-mail: leileicuiccnu@163.com; Jiaxing GUO, E-mail: 842365783@qq.com
  • Supported by:
    Natural Science Foundation of China (11771166, 12071169), the Hubei Key Laboratory ofMathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University # IRT17R46.

Abstract: In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations
(ε2a+εbR3|u|2)Δu+V(x)u=up,u>0inR3,
which concentrate at non-degenerate critical points of the potential function V(x), where a,b>0, 1<p<5 are constants, and ε>0 is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at {ai}1ik, where {ai}1ik are non-degenerate critical points of V(x) as ε0.

Key words: Kirchhoff type equations, potential functions having non-degenerate critical points, the Lyapunov-Schmidt reduction method, multi-peak solutions, existence and local uniqueness

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