The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation i∂tu+Δu=λ|x|-α|u|βu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$\end{document} in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}. The well-posedness theory in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} has been intensively studied in recent years, but the currently known approaches do not work for the critical case β=(4-2α)/(n-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =(4-2\alpha )/(n-2)$$\end{document}. It is still an open problem. The main contribution of this paper is to develop the theory in this case.