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

Abstract

In this paper, we consider the following Brézis-Nirenberg problem with prescribed $L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -Δu=|u|^{2^*-2} u +λ_ρu\quad \text { in } Ω, u>0, \quad u \in H_0^1(Ω), \quad \int_Ω u^2dx=ρ, \end{cases} \end{equation*} where $N⩾6$, $2^{\ast }=2N/(N-2)$ 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 Brézis-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.