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We study a class of two-dimensional non-linear Schr\"odinger equations withpoint-like singular perturbation and Hartree non-linearity. The point-likesingular perturbation of the free Laplacian induces appropriate perturbedSobolev spaces that are necessary for the study of ground states and evolutionflow. We include in our treatment both mass sub-critical and mass criticalHartree non-linearities. Our analysis is two-fold: we establish existence,symmetry, and regularity of ground states, and we demonstrate thewell-posedness of the associated Cauchy problem in the singular perturbedenergy space. The first goal, unlike other treatments emerging in parallel withthe present work, is achieved by a non-trivial adaptation of the standardproperties of Schwartz symmetrisation for the modified Weinstein functional.This produces, among others, modified Gagliardo-Nirenberg type inequalitiesthat allow to efficiently control the non-linearity and obtain well-posednessby energy methods. The evolution flow is proved to be global in time in thedefocusing case, and in the focusing and mass sub-critical case. It is alsoglobal in the focusing and mass critical case, for initial data that aresuitably small in terms of the best Gagliardo-Nirenberg constant.

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction