Nonlinear Schr?dinger equation in the Bopp-Podolsky electrodynamics: Global boundedness, blow-up and no scattering in the energy space

This paper concerns the nonlinear Schrodinger equation in the Bopp-Podolsky electrodynamics. By con-sidering a minimization problem related to the virial identity, we prove the existence of the ground state. In our approach we use the linear profile decomposition to recover compactness, which is distinguished with the mostly used mountain pass theorem. By doing a variational estimate below the ground state, we give the dichotomy of global boundedness and blow-up for solutions with energy below the ground state. As a consequence of the dichotomy, we show the strong instability of the standing wave. In the last part, the non-existence of scattering state will be proved based on the latest result of Murphy-Nakanishi.(c) 2023 Elsevier Inc. All rights reserved.