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We consider nonlinear Schr\"odinger equation with a Hartree-type nonlocalnonlinearity. The case where a nonlinear interaction potential grows at thespatial infinity is studied. By virtue of an effective decomposition of thenonlinearity based on conservation of mass, this kind of growing nonlinearinteraction is known to contain an effect like a linear potential. In thispaper, a well-posedness result is obtained in a suitable energy space for aclass of interaction potential growing at the spatial infinity in at most thequadratic order. If the growth rate of interaction potential is faster than thelinear order, a priori information on the center of mass plays a crucial role.When the interaction potential is exactly in the quadratic order, solution arewritten explicitly.

Analysis of Hartree equation with an interaction growing at the spatial infinity