Nonexistence of scattering and modified scattering states for some nonlinear Schr\"odinger equation with critical homogeneous nonlinearity

TL;DR

This paper proves the nonexistence of scattering and modified scattering states for certain critical nonlinear Schrödinger equations with non-polynomial nonlinearities, highlighting the limitations of free solution asymptotics in these cases.

Contribution

It extends previous results by including non-oscillating nonlinearities and demonstrates nonexistence of asymptotic free solutions under broader conditions.

Findings

No solutions behave like free solutions with or without phase corrections.

Nonexistence of asymptotic free solutions when gauge invariant nonlinearity dominates.

Finite time blow-up occurs in certain cases.

Abstract

We consider large time behavior of solutions to the nonlinear Schr\"odinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.