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

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations: [Formula: see text] subject to the mass constraints [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i.e. [Formula: see text]. For some suitable small [Formula: see text], 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 [Formula: see text], 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. [Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems, Calc. Var. Partial Differential Equations 62(1) (2023), Paper No. 9, 34pp.].