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

Abstract

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 3$, $1<α$, $2^{\ast}=\frac{2N}{N-2}$ and $\{u_α\}\subset H_{0}^{1,2α}(Ω)$ be a critical point of the functional \begin{equation*} I_{α,λ}(u)=\frac{1}{2α}\int\limits_Ω [(1+|\nabla u|^2)^α-1 ]dx-\fracλ{2}\int\limits_Ωu^2dx-\frac{1}{2^{\ast}}\int\limits_Ω|u|^{2^{\ast}}dx. \end{equation*} In this paper, we obtain the limit behaviour of $u_α$ ( $α\rightarrow 1$), 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} &-Δu=|u|^{2^{\ast}-2}u+λu\ \ \ \mbox{in}\ Ω,\\ &u=0,\ \ \mbox{on}\ \partialΩ. \end{aligned} \right. \end{equation*}