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

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 below a threshold and monotone above a threshold. Compared to the existing literature, rigorous asymptotics are derived by constructing families of solutions to the stationary equation with functional-analytic rather than geometric methods. The same analytical technique allows us to characterize the Morse index of the ground state.