Singular solutions of the subcritical nonlinear Schrِdinger equation

We show that the subcritical d -dimensional nonlinear Schrödinger equation i ψ t + Δ ψ + | ψ | 2 σ ψ = 0 , where 1 < σ d < 2 , admits smooth solutions that become singular in  L p for p ∗ < p ≤ ∞ , where p ∗ ≔ σ d σ d − 1 . Since lim σ d → 2 − p ∗ = 2 , these solutions can collapse at any 2 < p ≤ ∞ , and in particular for p = 2 σ + 2 .