Concentration phenomena of positive solutions to weakly coupled Schr\"odinger systems with large exponents in dimension two

We study the weakly coupled nonlinear Schr\"odinger system \begin{equation*} \begin{cases} -\Delta u_1 = \mu_1 u_1^{p} +\beta u_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } \Omega,\\ -\Delta u_2 = \mu_2 u_2^{p} +\beta u_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} \text{ in } \Omega,\\ u_1,u_2>0\quad\text{in }\;\Omega;\quad u_1=u_2=0 \quad\text { on } \;\partial\Omega, \end{cases} \end{equation*} where $p>1, \mu_1, \mu_2, \beta>0$ and $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$. Under the natural condition that holds automatically for all positive solutions in star-shaped domains \begin{align*} p\int_{\Omega}|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2 dx \leq C, \end{align*} we give a complete description of the concentration phenomena of positive solutions $(u_{1,p},u_{2,p})$ as $p\rightarrow+\infty$, including the $L^{\infty}$-norm quantization $\|u_{k,p}\|_{L^\infty(\Omega)}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p\int_{\Omega}|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2dx\to 8n\pi e$ with $n\in\mathbb{N}_{\geq 2}$, and so on. In particular, we show that the ``local mass'' contributed by each concentration point must be one of $\{(8\pi,8\pi), (8\pi,0),(0,8\pi)\}$.