  We study the stationary solutions of the Gross-Pitaevskii equation that
reduce, in the limit of vanishing non-linearity, to the eigenfunctions of the
associated Schr\"odinger equation. By providing analytical and numerical
support, we conjecture an existence condition for these solutions in terms of
the ratio between their proper frequency (chemical potential) and the
corresponding linear eigenvalue. We also give approximate expressions for the
stationary solutions which become exact in the opposite limit of strong
non-linearity. For one-dimensional systems these solutions have the form of a
chain of dark or bright solitons depending on the sign of the non-linearity. We
demonstrate that in the case of negative non-linearity (attractive interaction)
the norm of the solutions is always bounded for dimensions greater than one.
