Paper detail

On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in Sobolev spaces $H^s$, $s\geq0$. The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where $α=0$. The conventional approach does not work particularly for the critical cases $β=\frac{4-2α}{d-2s}$. It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted $L^p$ setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the singularity $|x|^{-α}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.