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In this paper we study the existence of positive normalized solutions of thefollowing coupled Schr\"{o}dinger system: \begin{align} \left\{ \begin{aligned}& -\Delta u = \lambda_u u + \mu_1 u^3 + \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 $\mu_1, \mu_2 > 0$,$\beta \neq 0$, $c_1, c_2 > 0$, and $\Omega \subset \mathbb{R}^N$ ($N = 3, 4$)is smooth, bounded, and star-shaped. Note that the nonlinearities and thecoupling terms are both $L^2$-supercritical in dimensions 3 and 4, Sobolevsubcritical in dimension 3, Sobolev critical in dimension 4. We show that thissystem has a positive normalized solution which is a local minimizer. Wefurther show that the system has a second positive normalized solution, whichis of M-P type when $N = 3$. This seems to be the first existence result of twopositive normalized solutions for such a Schr\"{o}dinger system, especially inthe Sobolev critical case. We also study the limit behavior of the positivenormalized solutions in the repulsive case $\beta \to -\infty$, and phaseseparation is expected.

Two Positive Normalized Solutions and Phase Separation for Coupled  Schr\"odinger Equations on Bounded Domain with L2-Supercritical and Sobolev  Critical or Subcritical Exponent