On Brezis-Nirenberg problems: Open questions and new results in dimension six
Abstract
In this paper, we consider the Brezis-Nirenberg problem$ \begin{equation*} \left\{\begin{aligned} &-\Delta u = \lambda u+|u|^{2^*-2}u, \quad &\text{in}\, \Omega, \\ &u = 0, \quad &\text{on}\, \partial\Omega, \end{aligned}\right. \end{equation*} $where $ \Omega $ is a smoothly bounded domain of $ \mathbb R^N $ with $ N\geq 3 $, $ \lambda>0 $ is a parameter, and $ 2^* = \frac{2N}{N-2} $ is the critical Sobolev exponent. First, we 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.
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Geometric Analysis and Curvature Flows
Stability and Controllability of Differential Equations
Publication details
- Journal / source
- Discrete and Continuous Dynamical Systems
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 54
- Issue
- 0
- Pages
- 151–176
- Publication date
- 1 January 2026