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

In this paper we study the existence of positive normalized solutions of the following 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 the coupling terms are both $L^2$-supercritical in dimensions 3 and 4, Sobolev subcritical in dimension 3, Sobolev critical in dimension 4. We show that this system has a positive normalized solution which is a local minimizer. We further show that the system has a second positive normalized solution, which is of M-P type when $N = 3$. This seems to be the first existence result of two positive normalized solutions for such a Schr\"{o}dinger system, especially in the Sobolev critical case. We also study the limit behavior of the positive normalized solutions in the repulsive case $\beta \to -\infty$, and phase separation is expected.