On Brezis-Nirenberg problems: open questions and new results in dimension six

2025/09/24 by Fengliu Li, Li, Fengliu, Giusi Vaira +5 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.19863

openalex publication_date 2025/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the Brezis-Nirenberg problem \\beginaligned amp;-Δu = λu+|u|^{2^*-2}u, amp;in Ω,
amp;u=0, amp;on ∂Ω, \endaligned. where Ω is a smoothly bounded domain of \mathbb R^N with N≥ 3, λ>0 is a parameter and 2^*=(2N)/(N-2) is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenberg problem.

Citations

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• On Brezis' First Open Problem: A Complete Solution
• Compactness results for Sign-Changing Solutions of critical nonlinear elliptic equations of low energy
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• Towers of bubbles for Yamabe-type equations and for the Brézis-Nirenberg problem in dimensions n ≥ 7
• Multispike solutions for the Brezis-Nirenberg problem in dimension three
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• Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem in low dimensions
• A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions
• Sign-changing tower of bubbles for the Brezis-Nirenberg problem
• Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem
• Sign-changing blow-up for scalar curvature type equations
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Cited by

• Existence of positive solutions for a class of almost critical problems on an annulus

Related

• A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions
• Sharp quantitative stability estimates for the Brezis-Nirenberg problem
• The Brezis-Nirenberg problem in 4D
• Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case
• Nodal solutions of the Brezis-Nirenberg problem in dimension 6