Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

TL;DR

This paper investigates positive, decaying solutions to the Gross-Pitaevskii equation with harmonic potential, detailing existence, asymptotic behavior, and Morse index of solutions in energy-critical and supercritical regimes across various dimensions.

Contribution

It provides a comprehensive analysis of the existence, asymptotics, and Morse index of solutions for energy-critical and supercritical nonlinearities in the Gross-Pitaevskii equation, including precise conditions and behaviors.

Findings

Existence of ground states depends on frequency range in different dimensions.

Asymptotic behavior of ground states is characterized up to leading order.

Morse index varies with nonlinearity power and oscillatory or monotone nature.

Abstract

We consider positive and spatially decaying solutions to the Gross-Pitaevskii equation with a harmonic potential. For the energy-critical case, there exists a ground state if and only if the frequency belongs to (1,3) in three dimensions and in (0,d) in d dimensions. We give a precise description on asymptotic behaviors of the ground state up to the leading order term for different values of d. For the energy-supercritical case, there exists a singular solution for some frequency in (0,d). We compute the Morse index of the singular solution in the class of radial functions and show that the Morse index is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for nonlinearity powers not large enough and is equal to 1 in the monotone case for nonlinearity power sufficiently large.