Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations $$\{\begin{array}{l}-\mathrm{\Delta }u+{\lambda }_{1}u={\mu }_1\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u+\nu \alpha \mathrm{\mid }u{\mathrm{\mid }}^{\alpha -2}\mathrm{\mid }v{\mathrm{\mid }}^{\beta }u\text{in}{\mathbb{R}}^N,\\ -\mathrm{\Delta }v+{\lambda }_{2}v={\mu }_2\mathrm{\mid }v{\mathrm{\mid }}^{2^{\ast }-2}v+\nu \beta \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^{\beta -2}v\text{in}{\mathbb{R}}^N\end{array}$$ subject to the mass constraints ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2\text{ud}x=a^2$ and ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }v{\mathrm{\mid }}^2\text{ud}x=b^2$, where, $a>0,b>0,N=3,4$ and $2^{\ast }:=\frac{2N}{N-2}$ is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., $\alpha >1,\beta >1,\alpha +\beta <2+\frac{4}{N}$. For some suitable small $\nu >0$, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as $\nu \to 0^{+}$, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave’s type open problem raised by Bartsch {\it et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023).