Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1141-1150.doi: 10.1007/s10473-021-0408-6

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MULTIPLE SOLUTIONS OF SOME ELLIPTIC SYSTEMS WITH LINEAR COUPLINGS

Yutong CHEN1, Jiabao SU1, Mingzheng SUN2, Rushun TIAN3   

  1. 1. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;
    2. College of Sciences, North China University of Technology, Beijing 100144, China;
    3. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
  • Received:2020-01-29 Revised:2020-10-30 Online:2021-08-25 Published:2021-09-01
  • Contact: Rushun TIAN E-mail:rushun.tian@cnu.edu.cn
  • Supported by:
    Supported by KZ202010028048, NSFC (12001382, 11771302, 11601353) and Beijing Education Committee (KM201710009012, 6943).

Abstract: In this paper, we study the existence of nontrivial solutions to the elliptic system \begin{equation*} \begin{cases} -\Delta u=\lambda v + F_u(x,u,v),& \ x\in\Omega,\\ -\Delta v=\lambda u + F_v(x,u,v),& \ x\in\Omega,\\ u=v=0,& \ x\in\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is bounded with a smooth boundary. By the Morse theory and the Gromoll-Meyer pair, we obtain multiple nontrivial vector solutions to this system.

Key words: Morse theory, multiplicity, elliptic system, linear couplings

CLC Number: 

  • 37B30
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