A note on the $H^{s}$-critical inhomogeneous nonlinear Schr\"{o}dinger equation

In this paper, we consider the Cauchy problem for the $H^{s}$-critical inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $$iu_{t} +\Delta u=\lambda |x|^{-b} f(u),\; u(0)=u_{0} \in H^{s} (\mathbb R^{n}),$$ where $n\in \mathbb N$, $0\le s<\frac{n}{2}$, $0<b<\min \left\{2,\;n-s,\; 1+\frac{n-2s}{2} \right\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda |u|^{\sigma } u$ with $\lambda \in \mathbb C$ and $\sigma=\frac{4-2b}{n-2s}$. First, we establish the local well-posedness as well as the small data global well-posedness in $H^{s}(\mathbb R^{n})$ for the $H^{s}$-critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, we obtain some standard continuous dependence results for the $H^{s}$-critical INLS equation. Our results about the well-posedness and standard continuous dependence for the $H^{s}$-critical INLS equation improve the ones of Aloui-Tayachi [Discrete Contin. Dyn. Syst. 41 (11) (2021), 5409-5437] by extending the validity of $s$ and $b$. Based on the local well-posedness in $H^{1}(\mathbb R^{n})$, we finally establish the blow-up criteria for $H^{1}$-solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.