  From the mathematical side, nonlinear Schr\"odinger equations are usually
investigated via variational methods, that cease to work in higher dimensions.
This thesis tries to overcome this problem by focusing on spherically symmetric
solutions. Then, one can use classical methods coming from the fields of
ordinary differential equations and dynamical systems. The results presented
here include existence and uniqueness of the stationary solutions, their
frequency, and stability. The dynamical properties of the resonant
approximation are also explored. The main focus is given to the
Schr\"odinger-Newton-Hooke equations that is shown to be a nonrelativistic
limit of perturbations of the anti-de Sitter spacetime.
