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

We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential: −Δu+|x|2u=ωu+|u|p−2uin Rd, where d⩾3 , p > 2 and ω > 0. For p=2dd−2 (energy-critical case) there exists a ground state u ω if and only if ω∈(ω∗,d) , where ω∗=1 for d = 3 and ω∗=0 for d⩾4 . We give a precise description on asymptotic behaviours of u ω as ω→ω∗ up to the leading order term for different values of d⩾3 . When p>2dd−2 (energy-supercritical case) there exists a singular solution u∞ for some ω∈(0,d) . We compute the Morse index of u∞ in the class of radial functions and show that the Morse index of u∞ is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.