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The energy super-critical Gross--Pitaevskii equation with a harmonicpotential is revisited in the particular case of cubic focusing nonlinearityand dimension d > 4. In order to prove the existence of a ground state (apositive, radially symmetric solution in the energy space), we develop theshooting method and deal with a one-parameter family of classical solutions toan initial-value problem for the stationary equation. We prove that thesolution curve (the graph of the eigenvalue parameter versus the supremum) isoscillatory for d <= 12 and monotone for d >= 13. Compared to the existingliterature, rigorous asymptotics are derived by constructing three families ofsolutions to the stationary equation with functional-analytic rather thangeometric methods.

Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential