Two Positive Normalized Solutions and Phase Separation for Coupled Schrödinger 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 Schrödinger system: \begin{align} \left{ \begin{aligned} & -Δu = λ_u u + μ_1 u^3 + βuv^2, \quad x \in Ω, \ & -Δv = λ_v v + μ_2 v^3 + βu^2 v, \quad x \in Ω, \ & u > 0, v > 0 \quad $\text{in }$ Ω, \quad u = v = 0 \quad $\text{on }$ \partialΩ, \end{aligned} \right. \nonumber \end{align} with the $L^2$ constraint \begin{align} \int_Ω|u|^2dx = c_1, \quad \quad \int_Ω|v|^2dx = c_2, \nonumber \end{align} where ${\mu }_1,{\mu }_2>0$, $\beta \mathrm{\ne }0$, $c_1,c_2>0$, and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N=3,4$) 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 Schrödinger system, especially in the Sobolev critical case. We also study the limit behavior of the positive normalized solutions in the repulsive case $\beta \to -\mathrm{\infty }$, and phase separation is expected.