D.E. Pelinovsky, J. Wei, and Y. Wu
Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities
Nonlinearity 36 (2023) 3684-3709
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>=4 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 the nonlinearity powers not large enough and is
equal to 1 in the monotone case for the nonlinearity powers sufficiently large.
Keywords:
Gross-Pitaevskii equation, energy critical and super-critical nonlinearity, positive solutions,
asymptotic behavior, Morse index.