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\\beta uv^2, \\quad x \\in \\Omega, \\\\ & -\\Delta v = \\lambda_v v + \\mu_2 v^3 + \\beta u^2 v, \\quad x \\in \\Omega, \\\\ & u > 0, v > 0 \\quad \\text{in } \\Omega, \\quad u = v = 0 \\quad \\text{on } \\partial\\Omega, \\end{aligned} \\right. \\nonumber \\end{align} with the $L^2$ constraint \\begin{align} \\int_{\\Omega}|u|^2dx = c_1, \\quad \\quad \\int_{\\Omega}|v|^2dx = c_2, \\nonumber \\end{align} where $\\mu_1, \\mu_2 > 0$, 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