Existence and Uniqueness of Constraint Minimizers for the Planar Schrodinger-Poisson System with Logarithmic Potentials
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.
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