P. Bizon, F. Ficek, D.E. Pelinovsky, and S. Sobieszek
Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential
Nonlinear Analysis 210 (2021) 112358 (36 papers)
Abstract:
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) 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.
Keywords:
Gross-Pitaevskii equation, energy super-critical case, oscillatory behavior, mononote behavior,
shooting method, rigorous asymptotics.