Scattering and blow up for nonlinear Schr\"{o}dinger equation with the averaged nonlinearity

Abstract

We consider the 3-dimensional nonlinear Schr\"{o}dinger equation (NLS) with average nonlinearity. This is a limiting model of NLS with strong magnetic confinement and a generalized model of the resonant system of NLS with a partial harmonic oscillator in terms of nonlinear power. We provide a new proof for the conservation law of kinetic energy and remove the restriction on nonlinearity. Moreover, in the case of focusing, super-quintic, and sub-nonic, we construct a new ground-state solution and classify the behavior of the solutions below the ground state. We demonstrate a sharp threshold for scattering and blow-up.