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u_2 &= \\lambda_2 u_2 + \\mu_2 |u_2|^{p_2 -2}u_2 + \\beta r_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \\end{aligned} \\right. \\] under the constraint \\[ \\int_{\\mathbb{R}^N}|u_1|^2 \\, dx = a_1,\\quad \\int_{\\mathbb{R}^N}|u_2|^2 \\, dx = a_2. \\] Here $a_1, a_2 >0$ are prescribed, $\\mu_1, \\mu_2, \\beta>0$, and the frequencies 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