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

Abstract

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=\lambda |x{|}^{-\alpha }|u{|}^{\beta }u$ in Sobolev spaces $H^s$, $s\ge 0$. 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.

To appear in J. Differential Equations, 21 pages