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 $\ln \mathrm{\mid }x\mathrm{\mid }\ast u^2$ and a logarithmic external potential $V(x)=\ln (1+\mathrm{\mid }x{\mathrm{\mid }}^2)$, which can be described by the $L^2$-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold ${\rho }^{\ast }\in (0,\mathrm{\infty })$ such that constraint minimizers exist if and only if $0<\rho <{\rho }^{\ast }$. In particular, the local uniqueness of positive constraint minimizers as $\rho \nearrow {\rho }^{\ast }$ is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.