  The energy super-critical Gross--Pitaevskii equation with a harmonic
potential is revisited in the particular case of cubic focusing nonlinearity
and dimension d > 4. In order to prove the existence of a ground state (a
positive, radially symmetric solution in the energy space), we develop the
shooting method and deal with a one-parameter family of classical solutions to
an initial-value problem for the stationary equation. We prove that the
solution curve (the graph of the eigenvalue parameter versus the supremum) is
oscillatory for d <= 12 and monotone for d >= 13. Compared to the existing
literature, rigorous asymptotics are derived by constructing three families of
solutions to the stationary equation with functional-analytic rather than
geometric methods.
