A nonlinear Schrödinger equation with Coulomb potential

In this paper, we study the Cauchy problem for nonlinear Schrödinger equations with Coulomb potential $${\rm{i}}{\partial _t}u + \Delta u {K \over {\left| x \right|}}u = \lambda \right|^{p - 1}}u$$ $$1 < p \le 5\,\,{\rm{on}}\,\,{\mathbb{R}^3}$$ . Our results reveal influence of long range K∣x∣−1 on existence and scattering theories equations. particular, prove global when is attractive, i.e., K > 0, theory repulsive, ≤ 0. The argument based newly-established interaction Morawetz-type inequalities equivalence Sobolev norms Laplacian operator potential.