The Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation

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$H^1$\mathrm{.}Thewell-posednesstheoryin$H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $\beta =(4-2\alpha )\mathrm{/}(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.