Ground state in the energy supercritical GrossPitaevskii equation with a harmonic potential
Abstract
The energy supercritical GrossPitaevskii 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 oneparameter family of classical solutions to an initialvalue 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 functionalanalytic rather than geometric methods.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.04929
 arXiv:
 arXiv:2009.04929
 Bibcode:
 2020arXiv200904929B
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 42 pages