Singular solutions of the subcritical nonlinear Schrodinger equation

Abstract

We show that the subcritical $d$-dimensional nonlinear Schr\"odinger equation $i \psi_t + \Delta \psi + |\psi|^{2 \sigma} \psi = 0$, where $1<\sigma d<2$, admits smooth solutions that become singular in~$L^p$ for $p^*<p \le \infty$, where $p^*:=\frac{\sigma d}{\sigma d -1}$. Since $\lim_{\sigma d \to 2-} p^* = 2$, these solutions can collapse at any $2<p \le \infty$, and in particular for $p = 2 \sigma+2$.