Mass concentration and uniqueness of ground states for mass subcritical rotational nonlinear Schr\"{o}dinger equations

TL;DR

This paper studies the existence, behavior, and uniqueness of ground states for a mass subcritical rotational nonlinear Schrödinger equation, revealing conditions on rotational velocity and interaction strength that determine minimizer existence and uniqueness.

Contribution

It establishes the existence and non-existence of minimizers based on rotational velocity, and proves the uniqueness of large-amplitude ground states under certain conditions.

Findings

Minimizers exist for all interaction strengths when rotational velocity is below a critical value.

No minimizers exist when rotational velocity exceeds the critical value.

Large-amplitude minimizers are unique up to a phase shift for fixed parameters.

Abstract

This paper considers ground states of mass subcritical rotational nonlinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u+i\Omega(x^\perp\cdot\nabla u)=\mu u+\rho^{p-1}|u|^{p-1}u \,\ \text{in} \,\ \mathbb{R}^2, \end{equation*} where $V(x)$ is an external potential, $\Omega>0$ characterizes the rotational velocity of the trap $V(x)$, $1<p<3$ and $\rho>0$ describes the strength of the attractive interactions. It is shown that ground states of the above equation can be described equivalently by minimizers of the $L^2-$ constrained variational problem. We prove that minimizers exist for any $\rho\in(0,\infty)$ when $0<\Omega<\Omega^*$, where $0<\Omega^*:=\Omega^*(V)<\infty$ denotes the critical rotational velocity of $V(x)$. While $\Omega>\Omega^*$, there admits no minimizers for any $\rho\in(0,\infty)$. For fixed $0<\Omega<\Omega^*$, by using energy estimates and blow-up analysis, we also analyze the limit behavior of minimizers as $\rho\to\infty$. Finally, we prove that up to a constant phase, there exists a unique minimizer when $\rho>0$ is large enough and $\Omega\in(0,\Omega^*)$ is fixed.