The energy identity of Sacks-Uhlenbeck operator and infinitely many solutions for Brezis-Nirenberg problem

Let $\mathrm{\Omega }$ be a bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 3$, $1<\alpha$, $2^{\ast }=\frac{2N}{N-2}$ 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 $u_{\alpha }$ ( $\alpha \to 1$), energy identity, Pohozaev identity, some integral estimates, etc. And using these results, we prove infinitely many solutions for the following Brezis-Nirenberg problem for $N\ge 7$: \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*}