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 Schrödinger equations $$\{\begin{array}{l}-\Delta u+{\lambda }_{1}u={\mu }_1|u{|}^{2^{\ast }-2}u+\nu \alpha |u{|}^{\alpha -2}|v{|}^{\beta }uin{\mathbb{R}}^N,\\ -\Delta v+{\lambda }_{2}v={\mu }_2|v{|}^{2^{\ast }-2}v+\nu \beta |u{|}^{\alpha }|v{|}^{\beta -2}vin{\mathbb{R}}^N\end{array}$$ 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 $2^{\ast }:=\frac{2N}{N-2}$ 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).

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