On well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger equation in the critical case

Abstract

In this paper we study the well-posedness for the inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ 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 $\alpha=0$. The conventional approach does not work particularly for the critical cases $\beta=\frac{4-2\alpha}{d-2s}$. 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 $|x|^{-\alpha}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.