ENERGY SOLUTION TO A SCHRODINGER-POISSON SYSTEM IN THE TWO-DIMENSIONAL WHOLE SPACE

We consider the Cauchy problem of the two-dimensional Schrodinger-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.