Uniqueness of minimizers for the mass subcritical planar Schrödinger-Poisson system with logarithmic convolution potential

This paper is concerned with constraint minimizers of the $ L^{2} $-subcritical planar Schrödinger-Poisson system with a logarithmic convolution and an external potential $ V(x) $ satisfying some suitable conditions, which can be described by the constraint minimization problem. We analyze the limiting behavior of positive minimizers as $ \rho\to \infty $. Based on which, the uniqueness of positive minimizers is also proved for sufficiently large $ \rho>0 $.