Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

In the present paper, we study the normalized solutions with least energy to the following system: $$\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{1}u={\mu }_1\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\beta r_1\mathrm{\mid }u{\mathrm{\mid }}^{r_1-2}\mathrm{\mid }v{\mathrm{\mid }}^{r_2}u& \text{in}{\mathbb{R}}^N,\\ -\mathrm{\Delta }v+{\lambda }_{2}v={\mu }_2\mathrm{\mid }v{\mathrm{\mid }}^{q-2}v+\beta r_2\mathrm{\mid }u{\mathrm{\mid }}^{r_1}\mathrm{\mid }v{\mathrm{\mid }}^{r_2-2}v& \text{in}{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}{u}^2=a_1^2\text{and}{\int }_{{\mathbb{R}}^N}{v}^2=a_2^2,\end{array}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $\beta$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.