arXiv 2112.136332021-12-27math.AP

质量亚临界转动非线性薛定谔方程基态的质量集中和唯一性

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

Yongshuai Gao, Yong Luo

备注：: 34 pages

摘要：

本文考虑基态质量亚临界转动非线性薛定谔方程\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*}，其中$V(x)$是一个外势，$Ω>0$表征了陷阱$V(x)$的转动速度，$1<p<3$和$ρ>0$描述了吸引相互作用的强度.结果表明，上述方程的基态可以等价地描述的$L^2-$约束变分问题的极小。证明了当$0<Ω<Ω^*$，其中$0<Ω^*:=Ω^*(V)<\infty$表示$V(x)$的临界转速时，任何$ρ\in(0,\infty)$都存在极小元。虽然$Ω>Ω^*$，但不允许任何$ρ\in(0,\infty)$的极小化器。对于固定的$0<Ω<Ω^*$，通过能量估计和blow-up分析，我们还分析了极小元的极限行为为$ρ\to\infty$。最后，我们证明了当$ρ>0$足够大且$Ω\in(0,Ω^*)$固定时，直到一个常数相位，存在唯一的极小元。

英文摘要：

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, $Ω>0$ characterizes the rotational velocity of the trap $V(x)$, $1<p<3$ and $ρ>0$ 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 $ρ\in(0,\infty)$ when $0<Ω<Ω^*$, where $0<Ω^*:=Ω^*(V)<\infty$ denotes the critical rotational velocity of $V(x)$. While $Ω>Ω^*$, there admits no minimizers for any $ρ\in(0,\infty)$. For fixed $0<Ω<Ω^*$, by using energy estimates and blow-up analysis, we also analyze the limit behavior of minimizers as $ρ\to\infty$. Finally, we prove that up to a constant phase, there exists a unique minimizer when $ρ>0$ is large enough and $Ω\in(0,Ω^*)$ is fixed.

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