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

Abstract

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger 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.