2026/07/29 by Qing Guo, Chengxiang Zhang
Mathematics · #math.AP
arxiv created 2026/07/29 · arxiv updated 2026/07/30
We construct segregated bubbling solutions for a two-component critical Schrödinger system of Brezis--Nirenberg type in a smooth bounded domain Ω⊂\mathbb RN, N≥5, with any fixed competitive coupling β<0. Suppose that the Robin function has two distinct prescribed critical points, each satisfying a local degree condition. For every sufficiently small ε>0, the system admits a nonnegative weak solution with both components nontrivial. Each component has a single-bubble profile and concentrates at one of the prescribed points. Each component also vanishes identically in a ball centered at the other concentration point; after rescaling by the natural bubble length, the radius of this ball tends to infinity. The main obstruction is that p=N/(N-2)∈(1,2), so the gradient of the interaction potential (s,t)↦ |s|p|t|p is not differentiable when one component vanishes and the other is nonzero. Hence the usual global Lyapunov--Schmidt reduction cannot be applied directly. We first solve a nonlinear exterior problem variationally. The resulting dead cores remove the cross-component coupling from the inner localization regions, where a projected critical reduction can then be carried out. The remaining scale and center equations are solved by Brouwer degree theory.