The focusing energy-critical nonlinear Schr\"odinger system with power-type growth nonlinearities in the radial case

TL;DR

This paper investigates the behavior of a coupled energy-critical nonlinear Schrödinger system with power nonlinearities in three to five dimensions, establishing conditions for scattering or blow-up in the radial case.

Contribution

It proves the existence of ground states and characterizes the scattering versus blow-up dichotomy for the system in the radial setting.

Findings

Existence of ground state solutions via concentration-compactness.

Finite-time blow-up conditions established.

Scattering results proved using concentration-compactness and rigidity methods.

Abstract

This work is concerned with a coupled system of focusing nonlinear Schr\"odinger equations involving general power-type nonlinearities in the energy-critical setting for dimensions $3\leq d\leq 5$ in the radial setting. Our aim is to demonstrate the scattering versus blow-up dichotomy in the radial case. To achieve this, we first prove the existence of ground state solutions using the concentration-compactness method combined with variational techniques. We then establish finite-time blow-up through a convexity argument and prove scattering by applying the concentration-compactness and rigidity method.