Energy critical fourth-order Schr\"odinger equation system with power-type nonlinearities in the radial case

Abstract

In this paper, we study a system of focusing fourth-order Schr\"odinger equations in the energy-critical setting with radial initial data and general power-type nonlinearities. The main idea is to generalize the analysis of such systems: we first establish several hypotheses on the nonlinearities and prove their implications. These implications are then used to establish a local well-posedness result in $H^2(\mathbb{R}^d)$ and and to prove the existence of ground state solutions. Using a virial argument, we demonstrate a blow-up result for initial data with negative energy or with kinetic energy exceeding that of the ground state. Finally, employing the concentration-compactness/rigidity method, we prove a scattering result for solutions whose energy and kinetic energy are below those of the ground state.