Normalized solutions to mixed dispersion nonlinear Schrödinger system with coupled nonlinearity

Abstract

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 $\hat{I}{”}^{2}u=\hat{I}”(\hat{I}”u)$ is the biharmonic operator, $\hat{I}{\pm }_1$, $\hat{I}{\pm }_2$, ${\hat{I}}^2>0$, $r_1$, $r_2>1$, $N\ge 1$. $\ddot{I}{}^2$ stands for the prescribed mass, and $\hat{I}»\in \mathbb{R}$ 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 $\hat{I}{\pm }_1=\hat{I}{\pm }_2$, the ground state exists for all $\ddot{I}>0$ 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 $N\ge 2$, we obtain the existence of radial nontrivial mountain pass solution for sufficiently small $\ddot{I}>0$.