Ground state solution of critical Schrödinger equation with singular potential

In this paper, we consider the following Schrödinger equation with singular potential:\begin{document}$ \begin{equation*} \begin{aligned} -\Delta u + V(|x|)u = f(u),\ \, x\in \mathbb{R}^{N}, \end{aligned} \end{equation*} $\end{document}where \begin{document}$ N\geqslant 3 $\end{document} , \begin{document}$ V $\end{document} is a singular potential with parameter \begin{document}$ \alpha\in(0,2)\cup(2,\infty) $\end{document} , the nonlinearity \begin{document}$ f $\end{document} involving critical exponent. First, by using the refined Sobolev inequality, we establish a Lions-type theorem. Second, applying Lions-type theorem and variational methods, we show the existence of ground state solution for above equation. Our result partially extends the results in Badiale-Rolando [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006)], and Su-Wang-Willem [Commun. Contemp. Math. 9 (2007)].