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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1 u=\mu _1 u^{p-1}+\beta r_1 u^{r_1-1}v^{r_2}&{}\hbox {in}\quad \mathbb {R}^N,\\ -\Delta v+\lambda _2 v=\mu _2 v^{q-1}+\beta r_2 u^{r_1}v^{r_2-1}&{}\hbox {in}\quad \mathbb {R}^N,\\ 0<u,v\in H^1(\mathbb {R}^N),\quad 1\le N\le 4,&{} \end{array}\right. } \end{aligned}$$

satisfying the normalization

$$\begin{aligned} \int _{\mathbb {R}^N}u^2 \textrm{d}x=a \quad \text{ and } \quad \int _{\mathbb {R}^N}v^2 \textrm{d}x=b. \end{aligned}$$

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.,

$$\begin{aligned} 2+\frac{4}{N}<p,q,r_1+r_2<2^* \end{aligned}$$

with \(2^*=\frac{2N}{(N-2)_+}, 1\le N\le 4\) and give a partial affirmative answer to one open question in Bartsch et al. (J Math Pures Appl (9), 106(4):583–614, 2016). 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 \(\beta >0\).