In this paper we study the existence of positive normalized solutions of the following coupled Schrödinger system: \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u = \lambda _u u + \mu _1 u^3 + \beta uv^2, \quad x \in \Omega , \\&-\Delta v = \lambda _v v + \mu _2 v^3 + \beta u^2 v, \quad x \in \Omega , \\&u> 0, v > 0 \quad \text {in } \Omega , \quad u = v = 0 \quad \text {on } \partial \Omega , \end{aligned} \right. \end{aligned}\) with the \(L^2\) constraint \(\begin{aligned} \int _{\Omega }|u|^2dx = c_1, \quad \quad \int _{\Omega }|v|^2dx = c_2, \end{aligned}\) where \(\mu _1, \mu _2 > 0\) , \(\beta \ne 0\) , \(c_1, c_2 > 0\) , and \(\Omega \subset \mathbb {R}^N\) ( \(N = 3, 4\) ) is 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. It has been known 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 by studying the mountain pass geometry. 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 \(\beta \rightarrow -\infty \) , and phase separation is expected.
