论文标题
具有对数电势的平面Schrodinger-Poisson系统的约束最小化器的存在和独特性
Existence and Uniqueness of Constraint Minimizers for the Planar Schrodinger-Poisson System with Logarithmic Potentials
论文作者
论文摘要
在本文中，我们研究了平面schrödinger-poisson系统的约束最小化的$ u $，具有对数卷积潜在$ \ ln | x | x | x | \ ast u^2 $和一个对数的外部电位$ v（x）= \ ln（1+ | x | x | x |^2）$，由$ l^2 $ crcripation a描述了一个分数a的典范。我们证明，（0，\ infty）$中有一个阈值$ρ^* \，因此当且仅当$ 0 <ρ<ρ^* $时，约束最小化器存在。特别是，通过克服对数卷积潜力的签名改变属性和在对数外部电位的翻译下，分析了正面约束最小化的局部唯一性作为$ρ\附近ρ^*$。
In this paper, we study constraint minimizers $u$ of the planar Schrödinger-Poisson system with a logarithmic convolution potential $\ln |x|\ast u^2$ and a logarithmic external potential $V(x)=\ln (1+|x|^2)$, which can be described by the $L^2$-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold $ρ^* \in (0,\infty)$ such that constraint minimizers exist if and only if $0<ρ<ρ^*$. In particular, the local uniqueness of positive constraint minimizers as $ρ\nearrowρ^*$ is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.
