Normalized ground states for a coupled Schr\"odinger system: Mass super-critical case

Abstract

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 Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1 u^{p-1}+\beta r_1 u^{r_1-1}v^{r_2}\quad &\hbox{in}~\mathbb{R}^N,\\ -\Delta v+\lambda_2 v=\mu_2 v^{q-1}+\beta r_2 u^{r_1}v^{r_2-1}\quad &\hbox{in}~\mathbb{R}^N,\\ 0<u,v\in H^1(\mathbb{R}^N), \, 1\leq N\leq 4,& \end{cases} $$ satisfying the normalization $$ \int_{\mathbb{R}^N}u^2 \mathrm{d}x=a \quad \mbox{and} \quad \int_{\mathbb{R}^N}v^2 \mathrm{d}x=b.$$ 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., $$2+\frac{4}{N}<p,q,r_1+r_2<2^*$$ with $2^*$ being the Sobolev critical exponent, defined by $2^*:=+\infty$ for $N=1,2$ and $2^*:=\frac{2N}{N-2}$ 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 $\beta>0$.