A nonlinear Schrödinger equation with Coulomb potential

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