Standing waves of nonlinear Schr?dinger equations with the gauge field

摘要

We study standing waves for nonlinear Schr?dinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field A _μ in terms of complex scalar field φ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type.