Stationary problem related to the nonlinear Schrödinger equation on the unit ball

In this paper, we study the stability of standing waves for the nonlinear Schrödinger equation on the unit ball in R ^N with Dirichlet boundary condition. We generalize the result of Fibich and Merle (2001 Physica D 155 13258), which proves the orbital stability of the least-energy solution with the cubic power nonlinearity in two space dimension. We also obtain several results concerning the excited states in one space dimension. Specifically, we show the linear stability of the first three excited states and we give a proof of the orbital stability of the kth excited state, restricting ourselves to the perturbation of the same symmetry as the kth excited state. Finally, our numerical simulations on the stability of the kth excited state are presented.