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

The energy super-critical Gross–Pitaevskii equation with a harmonic potential is revisited in the particular case of cubic focusing nonlinearity and dimension d ≥ 5 . 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 norm) 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.