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 ⊂ 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.

中文翻译：

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一些带线性耦合的椭圆系统的多重解

在本文中，我们研究了椭圆系统的非平凡解的存在性

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

其中 Ω ⊂ ℝ ^N以平滑边界为界。通过 Morse 理论和 Gromoll-Meyer 对，我们获得了该系统的多个非平凡向量解。