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 cases -Δu+λ₁ u=μ₁|u|2^*-2u+να|u|α-2|v|^βu ~in~ N,\\ -Δv+λ₂ v=μ₂|v|2^*-2v+νβ|u|^α|v|β-2v ~in~ N cases subject to the mass constraints ∫RN|u|² x=a² and ∫RN|v|² x=b², where, $a>0,\,b>0,\,N=3,4$ and 2^*:=2N/N-2 is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., α>1,β>1,α+β<2+4/N. For some suitable small $ν>0$, 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 ν→0⁺, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch { et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023).