Existence and Uniqueness of Normalized Multi-peak Solutions for Coupled Nonlinear Schr\"odinger Systems

TL;DR

This paper proves the existence and local uniqueness of multi-peak solutions for a coupled nonlinear Schrödinger system under mass constraints, revealing new phenomena compared to unconstrained or single-peak solutions.

Contribution

It introduces a novel analysis combining Lyapunov-Schmidt reduction and Pohozaev identities to handle mass constraints in multi-peak solutions of CNLS systems.

Findings

Existence of multi-peak solutions for small mass in 3D.

Solutions exist near a critical mass threshold in 2D.

New phenomena differ from unconstrained and single-peak cases.

Abstract

We consider the following two-component coupled nonlinear Schr\"odinger (CNLS) system: \[ \begin{cases} -\Delta u +(P(x) + \lambda ) u=\mu_1 u^3+\beta u v^2, & \text{in } \mathbb{R}^N,\\ -\Delta v +(Q(x) + \lambda ) v =\mu_2 v^3+\beta vu^2, & \text{in } \mathbb{R}^N \end{cases} \] with the mass constraint $\int_{\mathbb{R}^N} (u^2+v^2)\,dx = \rho^2$ for $N=2,3$, where $\rho>0$ is a parameter. By employing the Lyapunov-Schmidt reduction and local Pohozaev identities, we establish the existence and local uniqueness of normalized multi-peak solutions: the result holds for sufficiently small $\rho$ when $N=3$, and for $\rho$ approaching a critical threshold when $N=2$. The main difficulty lies in that the mass constraint involves interactions among all concentration points, while a more refined characterization of such normalized solutions further requires sharp order estimates. In this work, we have discovered some new phenomena that differ from those of solutions without mass constraint and single-peak solutions.