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.

To appear in Arch. Math., 12 pages

References in corpus (2)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• On the GWP of focusing energy-ciritical inhomogeneous NLS

Cited by in corpus (2)

• Blow-up of radial solutions for the intercritical inhomogeneous NLS equation
• On the focusing energy-critical inhomogeneous NLS: weighted space approach