Normalized ground states for a coupled Schrödinger system: Mass super-critical case

We consider the existence of solutions $({\lambda }_1,{\lambda }_2,u,v)\in {\mathbb{R}}^2\times (H^1({\mathbb{R}}^N){)}^2$ to systems of coupled Schrödinger equations $$\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{1}u={\mu }_1{u}^{p-1}+\beta r_1{u}^{r_1-1}{v}^{r_2}& \text{in}{\mathbb{R}}^N,\\ -\mathrm{\Delta }v+{\lambda }_{2}v={\mu }_2{v}^{q-1}+\beta r_2{u}^{r_1}{v}^{r_2-1}& \text{in}{\mathbb{R}}^N,\\ 0<u,v\in H^1({\mathbb{R}}^N),1\le N\le 4,& \end{array}$$ satisfying the normalization $${\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=aand{\int }_{{\mathbb{R}}^N}{v}^2\mathrm{d}x=b\mathrm{.}$$ Here ${\mu }_1,{\mu }_2,\beta >0$ and the prescribed masses $a,b>0$. We focus on the coupled purely mass super-critical case, i.e., $$2+\frac{4}{N}<p,q,r_1+r_2<2^{\ast }$$ with $2^{\ast }$ being the Sobolev critical exponent, defined by $2^{\ast }:=+\mathrm{\infty }$ for $N=1,2$ and $2^{\ast }:=\frac{2N}{N-2}$ for $N=3,4$. We optimize the range of $(a,b,\beta ,r_1,r_2)$ for the existence. In particular, for $N=3,4$ with $r_1,r_2\in (1,2)$, our result indicates the existence for all $a,b>0$ and $\beta >0$.