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

Abstract

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 , p > 2 and ω > 0. For (energy-critical case) there exists a ground state u ω if and only if , where for d = 3 and for . We give a precise description on asymptotic behaviours of u ω as up to the leading order term for different values of . When (energy-supercritical case) there exists a singular solution for some . We compute the Morse index of in the class of radial functions and show that the Morse index of 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.