Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schrödinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$-\mathrm{\Delta }u+\lambda u=\mu \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }x{\mathrm{\mid }}^{-d}\mathrm{\mid }u{\mathrm{\mid }}^{2_d^{\ast }-2}uin{\mathbb{R}}^N,N\ge 3$$ where $\lambda \in \mathbb{R}$, $\mu >0$, $0<b,d<2$, $2+(4-2b)\mathrm{/}N<q<2+(4-2b)\mathrm{/}(N-2)$ and $2_d^{\ast }=2(N-d)\mathrm{/}(N-2)$ is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the equation with critical exponential growth \begin{equation}\nonumber \begin{aligned} &-Δu+λu=|x|^{-b}f(u) ;;\mbox{in};; $\mathbb{R}$^2 \end{aligned} \end{equation} where the nonlinearity $f(s)$ behaves like $\exp (s^2)$ as $s\to \mathrm{\infty }$. We extend the existence results due to Alves-Ji-Miyagaki (Calc. Var. 61, 2022) from $b=d=0$ to the case $0<b,d<2$.