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 $Δ^2u=Δ(Δu)$ is the biharmonic operator, $α_{1}$, $α_{2}$, $β>0$, $r_{1}$, $r_{2}>1$, $N\geq 1$. $ρ^2$ stands for the prescribed mass, and $λ\in\mathbb{R}$ 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 $α_1=α_2$, the ground state exists for all $ρ>0$ 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 $ρ>0$.