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We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poissonsystem in the energy class. Though the Newtonian potential diverges at thespatial infinity in the logarithmic order, global well-posedness is proven inboth defocusing and focusing cases. The key is a decomposition of thenonlinearity into a sum of the linear logarithmic potential and a goodremainder, which enables us to apply the perturbation method. Our argument canbe adapted to the one-dimensional problem.

Energy solution to Schr\"odinger-Poisson system in the two-dimensional whole space