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

Abstract: The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left{\begin{aligned} &-\delta u+\lambda_1 u=|u|^{2^*-2}u+{\nu\alpha} |u|^{\alpha-2}|v|^\beta u,\quad \text{in }\mathbb{R}^N, &-\delta v+\lambda_2 v=|v|^{2^*-2}v+{\nu\beta} |u|^\alpha |v|^{\beta-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha,\beta>1$, $2<\alpha+\beta<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $\nu<0$. When $\nu>0$ and $\alpha+\beta\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<\nu<\nu_0$, the constant $\nu_0$ will be explicitly given. In the case $\alpha+\beta>2+\frac{4}{N}$ we prove the existence of a threshold $\nu_1\ge 0$ such that a normalized ground state solution exists for $\nu>\nu_1$, and does not exist for $\nu<\nu_1$. We also give conditions for $\nu_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $\nu\to0^+$ or $\nu\to+\infty$.