Normalized solutions for nonlinear Schr\"odinger systems with special mass-mixed terms: The linear couple case

In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad &\hbox{in}\;\RN, \\ -\Delta v + \lambda_2 v= \mu_2|v|^{q-2}v+\beta u \quad &\hbox{in}\;\RN, \end{cases}$$ satisfying the normalization constraints $\displaystyle\int_{\RN}u^2 =a, ~ \int_{\RN}v^2 =b$. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed and the masses $a,b>0$. Here $2+\frac{4}{N}<p,q\leq 2^*$, where $2^* = \frac{2N}{N-2}$ if $N \geq 3$ and $2^* =+ \infty$ if $N=2$. So that the terms $\mu_1|u|^{p-2}u$,$\mu_2|v|^{q-2}v$ are of the so-called mass supercritical, while the linear couple terms $\beta v, \beta u$ are of mass subcritical. An essential novelty is that this is the first try to deal with the linear couples in the normalized solution frame with mass mixed terms, which are big nuisances due to the lack of compactness of the embedding $H^1(\R^N)\hookrightarrow L^2(\R^N)$, even working in the radial subspace. For the Sobolev subcritical case, we can obtain the existence of positive ground state solution for any given $a,b>0$ and $\beta>0$, provided $2\leqslant N\leqslant 4$. For the Sobolev critical case with $N=3,4$, it can be viewed as a counterpart of the Brezis-Nirenberg critical semilinear elliptic problem for the system case in the context of normalized solutions. Under some suitable assumptions, we obtain the existence or non-existence of positive normalized ground state solution.