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

2025/05/20 by Zongyan Lv, Lv, Zongyan, Xiaoyu Zeng +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.14168

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the following Brézis-Nirenberg problem with prescribed L²-norm (mass) constraint: \begincases -Δu=|u|^{2^*-2} u +λ_ρu \text in Ω, ugt;0, u ∈ H₀¹(Ω), ∫_Ω u²dx=ρ, \endcases 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∈ ℝ⁺, we construct a k-spike solutions in some suitable bounded domain Ω. Our results extend those in \citeBHG3,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.

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• Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case
• On Brezis-Nirenberg problems: open questions and new results in dimension six