Classification of solitary wave solutions to Schrödinger equations with potentials

Abstract

In this paper, we prove the existence and uniqueness of solitary wave solutions of time-dependent nonlinear Schrödinger equations with behaviors tending to zero at infinity under certain conditions on trapping potentials and parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. Particularly, for constant trapping potentials, we conclude that there is at most one radial ground state under certain conditions on parameters.