Positive normalized solutions to nonlinear elliptic systems in ${\mathbb{R}}^4$ with critical Sobolev exponent

In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta v^2u\quad&\hbox{in}~\R^4,\ -\Delta v+\lambda_2v=\mu_2v^3+\alpha_2|v|^{p-2}v+\beta u^2v\quad&\hbox{in}~\R^4,\ \end{cases} \end{equation} under the mass constraint $${\int }_{{\mathbb{R}}^4}{u}^2=a_1^2\text{and}{\int }_{{\mathbb{R}}^4}{v}^2=a_2^2,$$ where $a_1,a_2$ are prescribed, ${\mu }_1,{\mu }_2,\beta >0$; $\alpha_1,\alpha_2\in \R$,$p!\in! (2,4)$and$\lambda_1,\lambda_2!\in!\R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., ${\alpha }_i<0(i=1,2)$. Then turning to the case of ${\alpha }_i>0(i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr"{o}dinger systems with Sobolev critical exponent. When $3\le p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.