Existence of normalized ground states for a singular equation with Sobolev critical growth nonlinearity

We investigate the existence and non-existence of solutions to the following problem with Sobolev critical nonlinearity:$\begin{align} \begin{cases} -\Delta u+\frac{u}{r^{2}}-\lambda u -\kappa |u|^{q-2}u-|u|^{2^*-2}u = 0, \ \ x = (y,z)\in \mathbb{R}^{K}\times\mathbb{R}^{N-K},\\ \int_{\mathbb{R}^{N}}|u|^{2}dx = \alpha^{2}>0, \end{cases} \end{align}$where $ N\geq3 $, $ 2^* = \frac{2N}{N-2} $, $ \kappa\in \mathbb{R} $, and $ r = |y| = \left(x_{1}^{2}+x_{2}^{2}+\cdot\cdot\cdot+x_{K}^2\right)^{\frac{1}{2}} $. Here, $ \lambda $ is a Lagrange multiplier, which appears due to the prescribed mass constraint $ |u|_2 = \alpha $. $ \kappa |u|^{q-2}u $ is the perturbation term of the equation. $ V(x) = \frac{1}{r^{2}} $ is referred to as a singular potential, and the Sobolev critical equation with a singular potential seems to be rarely studied. However, this paper presents some results as follows. In the case of radial symmetry, i.e. $ N = K $, we explore the existence and non-existence of normalized ground states to the equation. Furthermore, we prove that the behavior of the ground states as $ \kappa\to0^{+} $.