Concentration phenomena of positive solutions to weakly coupled Schrödinger systems with large exponents in dimension two

We study the weakly coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -Δu_1 = μ_1 u_1^{p} +βu_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}$\text{ in }$ Ω,\ -Δu_2 = μ_2 u_2^{p} +βu_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} $\text{ in }$ Ω,\ u_1,u_2>0\quad$\text{in }$;Ω;\quad u_1=u_2=0 \quad\text { on } ;\partialΩ, \end{cases} \end{equation*} where $p>1,{\mu }_1,{\mu }_2,\beta >0$ and $\mathrm{\Omega }$ 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_Ω|\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 $p\to +\mathrm{\infty }$, including the $L^{\mathrm{\infty }}$-norm quantization $\mathrm{\parallel }{u}_{k,p}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{u}_{1,p}{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }{u}_{2,p}{\mathrm{\mid }}^{2}dx\to 8n\pi e$ with $n\in {\mathbb{N}}_{\ge 2}$, and so on. In particular, we show that the ``local mass’’ contributed by each concentration point must be one of $\{(8\pi ,8\pi ),(8\pi ,0),(0,8\pi )\}$.