  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.
