Energy Solution to a Schrödinger–Poisson System in the Two-Dimensional Whole Space

We consider the Cauchy problem of the two-dimensional Schrödinger–Poisson system in the energy class. Though the Newtonian potential diverges at spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem.