Segregated Bubbling Solutions for a Critical Schr\"odinger System of Brezis--Nirenberg Type with Sublinear Competitive Coupling

We construct segregated bubbling solutions for a two-component critical Schr"odinger system of Brezis–Nirenberg type in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 5$, with any fixed competitive coupling $\beta <0$. 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 $p=N\mathrm{/}(N-2)\in (1,2)$, so the gradient of the interaction potential $(s,t)\mapsto \mathrm{\mid }s{\mathrm{\mid }}^p\mathrm{\mid }t{\mathrm{\mid }}^p$ 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.