Existence and local uniqueness of multi-spike solutions for Br\'{e}zis-Nirenberg problem with prescribed mass

Abstract

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.