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

TL;DR

This paper analyzes the energy behavior of solutions to a Sacks-Uhlenbeck type operator and uses these insights to establish the existence of infinitely many solutions for the Brezis-Nirenberg problem in higher dimensions.

Contribution

It provides the limit behavior, energy identity, and integral estimates for critical points of a Sacks-Uhlenbeck functional, leading to new existence results for the Brezis-Nirenberg problem.

Findings

Limit behavior of solutions as alpha approaches 1.

Energy and Pohozaev identities established.

Infinitely many solutions for N ≥ 7.

Abstract

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*}