In this paper, we consider the existence of normalized solutions for the following biharmonic nonlinear Schr\"{ o}dinger system \begin{ equation*} \begin{ cases} \Delta^2u+\alpha_{ 1}\Delta u+\lambda u=\beta r_{ 1}|u|^{ r_{ 1}-2}|v|^{ r_{ 2}} u & \text { in } \mathbb{ R}^{ N}, \\ \Delta^2v+\alpha_{ 2}\Delta v+\lambda v=\beta r_{ 2}|u|^{ r_{ 1}}|v|^{ r_{ 2}-2} v & \text { in } \mathbb{ R}^{ N}, \\ \int_{ \mathbb{ R}^{ N}} (u^{ 2}+v^{ 2})\ud x=\rho^{ 2}, \end{ cases} \end{ equation*} where $\Delta^2u=\Delta(\Delta u)$ is the biharmonic operator, $\alpha_{ 1}$, $\alpha_{ 2}$, $\beta>0$, $r_{ 1}$, $r_{ 2}>1$, $N\geq 1$. $\rho^2$ 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 $r_1+r_2<\min\left\{ \max\left\{ 4, 2+\frac{ 8}{ N+1}\right\}, 2+\frac{ 8}{ N}\right\}$. When $r_{ 1}+r_{ 2}\in\left(2+\frac{ 8}{ N}, \frac{ 2N}{ (N-4)^{ +}}\right)$ and $N\geq 2$, we obtain the existence of radial nontrivial mountain pass solution for sufficiently small $\rho>0$.
Categories: math.AP