Normalized solutions to mixed dispersion nonlinear Schrödinger system with coupled nonlinearity
Zhen-Feng Jin, Guotao Wang, Weimin Zhang
TL;DR
This work analyzes normalized (mass-constrained) solutions to a biharmonic, two-component nonlinear Schrödinger system with mixed dispersion. By formulating a single mass constraint framework and employing variational methods, concentration-compactness, and radial symmetry, it establishes a dichotomy in the mass-subcritical/critical regime and proves the existence of radial mountain-pass states in the mass-supercritical range. A central tool is the comparison between the constrained energy $m(\rho)$ and the decoupled energy $m^{J}(\rho)$, governed by a critical threshold $\rho^{*}$ derived from a supremum $R$ of a coupling functional $Q(u,v)$. The results delineate when ground states exist and when radial excited states can be obtained, offering a detailed picture of how mixed dispersion and coupled nonlinearity shape the energy landscape. These findings advance the theory of normalized solutions for multi-component biharmonic systems and clarify the role of mass, exponents, and symmetry in the existence and character of solutions.
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 $ρ>0$.