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 ₑ₍|u|² x=a² and ₑ₍|v|² x=b², where, a>0, \, b>0, \, N=3, 4 and 2^*: =2NN-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+β0, 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 ν0^+, 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).
Zhang et al. (Thu,) studied this question.