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∂tu+Δu=λ|x|−α|u|βu in Sobolev spaces Hs, s≥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 α=0. The conventional approach does not work particularly for the critical case β=4−2αd−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 Lp 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 spatially decaying factor |x|−α in the nonlinearity more efficiently. This observation is a core of our approach that covers the critical case successfully.