  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.
