  Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 3$,
$1<\alpha$, $2^{\ast}=\frac{2N}{N-2}$ and $\{u_\alpha\}\subset
H_{0}^{1,2\alpha}(\Omega)$ be a critical point of the functional
\begin{equation*} I_{\alpha,\lambda}(u)=\frac{1}{2\alpha}\int\limits_{\Omega}
[(1+|\nabla u|^2)^{\alpha}-1
]dx-\frac{\lambda}{2}\int\limits_{\Omega}u^2dx-\frac{1}{2^{\ast}}\int\limits_{\Omega}|u|^{2^{\ast}}dx.
\end{equation*} In this paper, we obtain the limit behaviour of $u_\alpha$ (
$\alpha\rightarrow 1$), energy identity, Pohozaev identity, some integral
estimates, etc. And using these results, we prove infinitely many solutions for
the following Brezis-Nirenberg problem for $N\geq 7$: \begin{equation*} \left\{
\begin{aligned}
  &-\Delta u=|u|^{2^{\ast}-2}u+\lambda u\ \ \ \mbox{in}\ \Omega,\\ &u=0,\ \
\mbox{on}\ \partial\Omega. \end{aligned} \right. \end{equation*}
