D.E. Pelinovsky and S. Sobieszek
Ground state of the Gross-Pitaevskii equation with a harmonic potential in the energy-critical case
Asymptotic Analysis 139 (2024) 1–29
Abstract:
Ground state of the energy-critical Gross-Pitaevskii equation with a harmonic potential
can be constructed variationally. It exists in a finite interval of the eigenvalue parameter.
The supremum norm of the ground state vanishes at one end of this interval and diverges to
infinity at the other end. We explore the shooting method in the limit of large norm to prove
that the ground state is pointwise close to the Aubin-Talenti solution of the energy-critical wave
equation in near field and to the confluent hypergeometric function in far field. The shooting
method gives the precise dependence of the eigenvalue parameter versus the supremum norm.
Keywords:
Gross-Pitaevskii equation, energy critical case, algebraic soliton, confluent hypergeometric function,
shooting method, rigorous asymptotics.