  In this paper we study the Cauchy problem for the energy-critical
inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta
u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in $H^1$. The well-posedness theory in
$H^1$ has been intensively studied in recent years, but the currently known
approaches do not work for the critical case $\beta=(4-2\alpha)/(n-2)$. It is
still an open problem. The main contribution of this paper is to develop the
theory in this case.
