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 \( -\Delta u+\lambda u=\mu |x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\text{ in }\;\; \mathbb {R}^N,\, N\ge 3, \) where \(\lambda \in \mathbb {R}\) , \(\mu >0\) , \(0<b,d<2\) , \(2+(4-2b)/N<q<2+(4-2b)/(N-2)\) and \(2^*_{d}= 2(N-d)/(N-2)\) is the Hardy–Sobolev critical exponent, while for \(N=2\) , we investigate the equation with critical exponential growth \(\begin{aligned} \begin{aligned}&-\Delta u+\lambda u=|x|^{-b}f(u) \;\;\text{ in }\;\; \mathbb {R}^2, \end{aligned} \end{aligned}\) where the nonlinearity f(s) behaves like \(\exp (s^2)\) as \(s\rightarrow +\infty \) . We extend the existence results due to Alves–Ji–Miyagaki (Calc. Var. 61, 2022) from \(b =d= 0\) to the case \(0< b,d < 2\) .
