  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.
