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

Abstract

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{2^*-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{2^*-2}v+\nu\beta |u|^{\alpha}|v|^{\beta-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where, $a>0,\,b>0,\,N=3,4$ and $2^*:=\frac{2N}{N-2}$ 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 $\nu\to0^+$, 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).