The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity

Abstract

In this paper, we study the Cauchy problem for the 3D energy-critical inhomogeneous nonlinear Schr\"odinger equation(INLS) $$i\partial_{t}u+\Delta u=\pm|x|^{-\alpha}|u|^{4-2\alpha}u$$ with strong singularity $3/2\leq \alpha<2$. The well-posedness problem is well-understood for $0<\alpha<3/2$, but the case $3/2\leq \alpha<2$ has remained open so far. We address the local/small data global well-posedness result for $3/2\leq \alpha<11/6$ by improving the inhomogeneous Strichartz estimates on the weighted space.