In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = λ|x| −α |u| β 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 α = 0. The conventional approach does not work particularly for the critical caseThe main contribution of this paper is to develop the well-posedness theory in this critical case (as well as noncritical 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 |x| −α in the nonlinearity more efficiently. This observation is a core of our approach that covers the critical case successfully.2−α β u(λx, λ 2 t), with the initial data u λ,0 (x) = u λ (x, 0) for all λ > 0. We then easily see
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