Existence of a ground state and scattering for a nonlinear Schroedinger equation with critical growth

TL;DR

This paper investigates the existence of ground states and scattering behavior for an energy-critical focusing nonlinear Schrödinger equation with subcritical perturbations in four or more dimensions.

Contribution

It establishes the existence of ground states and provides a necessary and sufficient condition for scattering in higher dimensions.

Findings

Existence of a ground state in four or higher dimensions.

Characterization of scattering conditions for solutions.

Extension of Kenig-Merle type results to perturbed energy-critical NLS.

Abstract

We study the energy-critical focusing nonlinear Schr\"odinger equation with an energy- subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig-Merle [16].