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

Abstract

Let $\hat{I}\text{c◯}$ be a bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 3$, $1<\hat{I}\pm$, $2^{\ast }=\frac{2N}{N-2}$ and $\{u_{\hat{I}}\pm \}\subset H_0^{1,2\hat{I}\pm }(\hat{I}\text{c◯})$ 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 $u_{\hat{I}}\pm$ ( $\hat{I}\pm \to 1$), 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} &-Δu=|u|^{2^{\ast}-2}u+λu\ \ \ \mbox{in}\ Ω,\\ &u=0,\ \ \mbox{on}\ \partialΩ. \end{aligned} \right. \end{equation*}