Singular solutions of the subcritical nonlinear Schrödinger equation

where x = (x1, . . . , xd) ∈ Rd and ∆ = ∑d j=1 ∂xjxj is the Laplacian, has been the subject of intensive study, due to its role in various areas of physics, such as nonlinear optics and Bose–Einstein condensates (BEC). The NLS is called subcritical, critical, and supercritical if σd < 2, σd = 2, and σd > 2, respectively. It is well-known that in the critical and supercritical cases, the NLS (1) possesses solutions that become singular in a finite time in Lp for some finite p [1]. In this study we show that the subcritical NLS also admits solutions that become singular in Lp for some finite p. NLS theory was originally developed for solutions that are in H1(Rd). In this case, the initial condition ψ0 ∈ H1, and the NLS solution is said to become singular at t = Tc if ψ(t) ∈ H1 for 0 ≤ t < Tc , and limt→Tc ‖ψ(t)‖H1 = ∞. In 1983, Weinstein proved that all H1 solutions of the subcritical NLS exist globally: