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

Abstract

Abstract We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential: − Δ u + | x | 2 u = ω u + | u | p − 2 u in  R d , where d ⩾ 3 , p > 2 and ω > 0. For p = 2 d d − 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 > 2 d d − 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.