The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential

Abstract

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge 3$, $\lambda =\pm 1$, $0<b<2$, $\sigma =\frac{4-2b}{d-2}$ and $c>-c(d):=-{\left(\frac{d-2}{2}\right)}^2$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^1$ for $c>-\frac{(d+2-2b{)}^2-4}{(d+2-2b{)}^2}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strichartz estimates. Based on the local well-posedness result, we then establish the blowup criteria for solutions to the equation in the focusing case $\lambda =-1$. To this end, we derive the sharp Hardy-Sobolev inequality and virial estimates related to this equation.

16 pages

References in corpus (1)

• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities