In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range pot...In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger 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≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.展开更多
基金The authors were supported by NSFC(12126409,12026407,11831004)the J.Zheng was also supported by Beijing Natural Science Foundation(1222019)。
文摘In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■_(t)u+△u+k/|x|u=λ/|u|^(p-l)u with 1<p≤5 on R^(3).Our results reveal the influence of the long range potential K|x|^(-1)on the existence and scattering theories for nonlinear Schrodinger 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≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.
