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Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $α,β>1$, $2<α+β<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $ν<0$. When $ν>0$ and $α+β\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<ν<ν_0$, the constant $ν_0$ will be explicitly given. In the case $α+β>2+\frac{4}{N}$ we prove the existence of a threshold $ν_1\ge 0$ such that a normalized ground state solution exists for $ν>ν_1$, and does not exist for $ν<ν_1$. We also give conditions for $ν_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $ν\to0^+$ or $ν\to+\infty$.