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Let $K$ be an imaginary quadratic field where $p$ is inert. Let $E$ be anelliptic curve defined over $K$ and suppose that $E$ has good supersingularreduction at $p$. In this paper, we prove that the plus/minus Selmer group of$E$ over the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ has nonon-trivial $\Lambda$-submodules of finite index under mild assumptions for$E$. This is an analogous result to R. Greenberg and B. D. Kim for theanticyclotomic $\mathbb{Z}_{p}$-extension essentially. By applying the resultsof A. Agboola--B. Howard or A. Burungale--K. B\"uy\"ukboduk--A. Lei, we canalso construct examples satisfying the assumptions of our theorem.

On non-trivial $\Lambda$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes