Mass concentration and uniqueness of ground states for mass subcritical rotational nonlinear Schrödinger equations

Abstract

This paper considers ground states of mass subcritical rotational nonlinear Schrödinger equation \begin{equation*} -Δu+V(x)u+iΩ(x^\perp\cdot\nabla u)=μu+ρ^{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 $0<\Omega <{\Omega }^{\ast }$, where $0<{\Omega }^{\ast }:={\Omega }^{\ast }(V)<\infty$ denotes the critical rotational velocity of $V(x)$. While $\Omega >{\Omega }^{\ast }$, there admits no minimizers for any $\rho \in (0,\infty )$. For fixed $0<\Omega <{\Omega }^{\ast }$, 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 $\Omega \in (0,{\Omega }^{\ast })$ is fixed.

34 pages