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Multiple Solutions of Some Elliptic Systems with Linear Couplings

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Abstract

In this paper, we study the existence of nontrivial solutions to the elliptic system

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = \lambda v + {F_u}(x,u,v),}&{x \in \Omega \;} \\ { - \Delta v = \lambda u + {F_v}(x,u,v),}&{x \in \Omega \;} \\ {u = u = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \partial \Omega } \end{array}} \right.$$

where Ω ⊂ ℝN is bounded with a smooth boundary. By the Morse theory and the Gromoll-Meyer pair, we obtain multiple nontrivial vector solutions to this system.

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Acknowledgements

The authors would like to thank the referee for giving valuable suggestions and a kind reminder of reference [4].

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Yutong Chen  (陈宇彤), Jiabao Su  (苏加宝) & Rushun Tian  (田如顺)

  2. College of Sciences, North China University of Technology, Beijing, 100144, China

    Mingzheng Sun  (孙明正)

Authors
  1. Yutong Chen  (陈宇彤)
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  2. Jiabao Su  (苏加宝)
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  3. Mingzheng Sun  (孙明正)
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  4. Rushun Tian  (田如顺)
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Corresponding author

Correspondence to Rushun Tian  (田如顺).

Additional information

Supported by KZ202010028048, NSFC (12001382, 11771302, 11601353) and Beijing Education Committee (KM201710009012, 6943).

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Chen, Y., Su, J., Sun, M. et al. Multiple Solutions of Some Elliptic Systems with Linear Couplings. Acta Math Sci 41, 1141–1150 (2021). https://doi.org/10.1007/s10473-021-0408-6

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  • DOI: https://doi.org/10.1007/s10473-021-0408-6

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