Threshold solutions for the $3d$ cubic INLS: the energy-critical case

Abstract

We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i{\partial }_{t}u+\hat{I}”u+|x{|}^{-1}|u{|}^{2}u=0$. In this work, we prove the existence of special solutions $W^{\pm }$ with energy equal to that of the ground state $W$ and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor $|x{|}^{-1}$ in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis.