%A Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG %T A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL %0 Journal Article %D 2022 %J Acta mathematica scientia,Series B %R 10.1007/s10473-022-0606-x %P 2230-2256 %V 42 %N 6 %U {http://121.43.60.238/sxwlxbB/CN/abstract/article_16863.shtml} %8 2022-12-25 %X In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential i$?_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with 1<p≤5 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 $K\leq0$. 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.