Infinitely many solutions to linearly coupled Schrödinger equations with non-symmetric potential

We study a linearly coupled Schrödinger system in $\R^N(N\leq3).$ Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combing localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends the result Theorem 1.2 about nonlinearly coupled Schrödinger equations in \cite{aw} to our linearly coupled problem.

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