Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials

Abstract

In this paper, we study constraint minimizers u of the planar Schrödinger-Poisson system with a logarithmic convolution potential ln⁡|x|⁎u2 and a logarithmic external potential V(x)=ln⁡(1+|x|2), which can be described by the L2-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold ρ⁎∈(0,∞) such that constraint minimizers exist if and only if 0<ρ<ρ⁎. In particular, the local uniqueness of positive constraint minimizers as ρ↗ρ⁎ is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.