四维Brezis-Nirenberg问题的变号多泡解
Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions
AI总结：
该研究针对四维Brezis-Nirenberg问题构造了变号多泡解，通过Lyapunov-Schmidt约化分析其性质，提出抽象存在性准则并应用于多种对称构型，还证明了双峰解的节点域相关性质。
AI中文摘要：
我们针对四维Brezis-Nirenberg问题构造了变号解族，当ε→0⁺时，问题形式为：在Ω内满足-Δu=u³+εu，在边界∂Ω上满足u=0。通过Lyapunov-Schmidt约化，证明了泡的位置和相对尺度由带符号的Green-Robin相互作用矩阵决定。我们基于存在一个简单正特征值（对应正特征向量）和一个稳定临界集，提出了一个抽象存在性准则，并将其应用于一般区域内的正负配对解，以及多种对称多峰构型，包括交替正多边形、正交多边形、一个中心峰被反号峰包围的构型，还有对齐的三峰、四峰和五峰模式。对于双峰解，我们还证明其恰好有两个节点域，且在自然平衡条件和边界连通性下，其节点集的闭包与边界相交。
英文摘要：
We construct families of sign-changing solutions for the four-dimensional Brezis--Nirenberg problem \[ -Δu=u^3+\varepsilon u\quad\text{in }Ω,\qquad u=0\quad\text{on }\partialΩ, \] as $\varepsilon\to0^+$. A Lyapunov--Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green--Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive--negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.
