Ground state in the energy super-critical Gross–Pitaevskii equation with a harmonic potential. (September 2021)

• Record Type: Journal Article
• Authors: Bizon, Piotr; Ficek, Filip; Pelinovsky, Dmitry E.; Sobieszek, Szymon
• Is Part Of: Nonlinear analysis. Volume 210(2021)
• Publication Date: 2021-09
• Subjects: Gross–Pitaevskii equation -- Ground states -- Emden–Fowler transformation -- Shooting method -- Solution curve; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112358
• Languages: English
• ISSNs: 0362-546X

Abstract

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 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.