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

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrodinger equation i partial derivative(t)u + Delta u = lambda vertical bar x vertical bar vertical bar(-alpha)vertical bar u vertical bar(beta)u in Sobolev spaces H-s, 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 alpha = 0. The conventional approach does not work particularly for the critical case beta = 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 spatially decaying factor vertical bar x vertical bar(-alpha) in the nonlinearity more efficiently. This observation is a core of our approach that covers the critical case successfully. (C) 2021 Elsevier Inc. All rights reserved.