Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass
Zongyan Lv, Xiaoyu Zeng, Huan-Song Zhou
TL;DR
This work addresses the existence, multiplicity, and local uniqueness of $L^2$-normalized solutions to the Brézis-Nirenberg problem with critical nonlinearity in a bounded domain for $N≥6$. It employs a Lyapunov-Schmidt reduction around the bubble family $U_{x,λ}$ to construct $k$-spike normalized solutions and a reduced functional $Ψ_k$ on an admissible set $𝒬_{Ω,k}$, requiring $M_k(a^k)$ to be positive and $(a^k,μ^k)$ a nondegenerate critical point of $Ψ_k$. The main results prove existence of normalized $k$-spike solutions for small mass $ρ$, with precise spike locations and scaling: $u_ρ ≈ extstyle\sum_{j=1}^k PU_{x_{j,ρ},μ_{j,ρ}}$ and $| abla u_ρ|^2 ightharpoonup frac{N}{2}\mathcal{S}^{N/2}\sum_{i=1}^k δ_{a_i}$ as $ρ→0^+$, together with the limits $ρ^{1/2}μ_{j,ρ}→(∫_{b R^N}U_{0,1}^2∑ μ_i^{-2})^{1/2} μ_j$ and $ρ^{(4-N)/2}λ_ρ→(∫_{b R^N}U_{0,1}^2∑ μ_i^{-2})^{(4-N)/2}$. It also establishes local uniqueness: two such solutions with the same $(a^k,μ^k)$ coincide for small $ρ$, and, under nondegeneracy for all admissible $μ^k$, the count of solutions equals $ ext{card}(𝒬_{a^k})$. This extends BN theory to mass-constrained, multi-peak configurations and provides a precise asymptotic and counting framework via blow-up analysis and local Pohozaev identities.
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 $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $ρ>0$ is a given small constant and $λ_ρ>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $Ω$. 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.