Normalized solutions to mixed dispersion nonlinear Schrödinger system with coupled nonlinearity

In this paper, we consider the existence of normalized solutions for the following biharmonic nonlinear Schrödinger system \begin{equation*} \begin{cases} Δ^2u+α_{1}Δu+λu=βr_{1}|u|^{r_{1}-2}|v|^{r_{2}} u & \text { in } $\mathbb{R}$^{N}, \ Δ^2v+α_{2}Δv+λv=βr_{2}|u|^{r_{1}}|v|^{r_{2}-2} v & \text { in } $\mathbb{R}$^{N}, \ \int_{$\mathbb{R}$^{N}} (u^{2}+v^{2})\ud x=ρ^{2}, \end{cases} \end{equation*} where ${\mathrm{\Delta }}^{2}u=\mathrm{\Delta }(\mathrm{\Delta }u)$ is the biharmonic operator, ${\alpha }_1$, ${\alpha }_2$, $\beta >0$, $r_1$, $r_2>1$, $N\ge 1$. ${\rho }^2$ stands for the prescribed mass, and $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier. Such single constraint permits mass transformation in two materials. When $r_1+r_2\le 2+\frac{8}{N}$, we obtain a dichotomy result with respect to the mass for the existence of nontrivial ground states. Especially when ${\alpha }_1={\alpha }_2$, the ground state exists for all $\rho >0$ if and only if $r_1+r_2<\min \{\max \{4,2+\frac{8}{N+1}\},2+\frac{8}{N}\}$. When $r_1+r_2\in (2+\frac{8}{N},\frac{2N}{(N-4{)}^{+}})$ and $N\ge 2$, we obtain the existence of radial nontrivial mountain pass solution for sufficiently small $\rho >0$.