标题： 具有预定质量的Brézis-Nirenberg问题的多尖点解的存在性和局部唯一性 显示英文标题

标题： Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass

摘要： 在本文中，我们考虑具有预定$ L^2$-范数（质量）约束的以下 Brézis-Nirenberg 问题： \begin{equation*} \begin{cases} -\Delta u=|u|^{2^*-2} u +\lambda_\rho u\quad \text { in } \Omega, u>0, \quad u \in H_0^1(\Omega), \quad \int_{\Omega} u^2dx=\rho, \end{cases} \end{equation*} 其中 $N \geqslant 6$， $2^*=2 N /(N-2)$是临界 Sobolev 指数， $\rho>0$是一个给定的小常数，而 $\lambda_\rho>0$作为 Euler-Lagrange 乘子。 对于任何$k\in \mathbb{R}^+$，我们在某些合适的有界域$\Omega$中构造了$k$-尖峰解。我们的结果扩展了\cite{BHG3,DGY,SZ}中的结果，其中作者获得了对应于上述方程的能量泛函的（局部）极小值或山路型临界点的一个或两个正解。此外，使用爆破分析和局部Pohozaev恒等式论证，我们证明了$k$-尖峰解是局部唯一的。与没有质量约束的标准Brézis-Nirenberg问题相比，在估计由不同解对应的欧拉-拉格朗日乘子差异引起的误差时出现了一个额外的困难。我们通过引入与线性化算子核相关的创新观察和估计来克服这一困难。 显示英文摘要

摘要： In this paper, we consider the following Br\'{e}zis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -\Delta u=|u|^{2^*-2} u +\lambda_\rho u\quad \text { in } \Omega, u>0, \quad u \in H_0^1(\Omega), \quad \int_{\Omega} u^2dx=\rho, \end{cases} \end{equation*} where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $\rho>0$ is a given small constant and $\lambda_\rho>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $\Omega$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Br\'{e}zis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators.