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

TL;DR

This paper investigates the existence and properties of ground states in the energy super-critical Gross-Pitaevskii equation with harmonic potential, focusing on cubic focusing nonlinearity in higher dimensions.

Contribution

It introduces a novel shooting method and provides rigorous asymptotics for ground states, revealing oscillatory and monotonic behaviors depending on the dimension.

Findings

Ground states exist for the specified equation in high dimensions.

Solution curves exhibit oscillatory behavior for dimensions ≤ 12.

Solution curves are monotone for dimensions ≥ 13.

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 > 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.