具次线性竞争耦合的Brezis--Nirenberg型临界薛定谔系统的分块气泡解
Segregated Bubbling Solutions for a Critical Schrödinger System of Brezis--Nirenberg Type with Sublinear Competitive Coupling
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中文总结 AI 辅助
针对N≥5的两分量Brezis--Nirenberg型临界薛定谔系统，通过变分法与Brouwer度理论，构造了具次线性竞争耦合的分块气泡解，克服了相互作用势梯度不可微的障碍。
中文摘要 AI 辅助
我们在光滑有界域Ω⊂ℝ^N（N≥5）中，对任意固定的竞争耦合β<0，构造了两分量Brezis--Nirenberg型临界薛定谔系统的分块气泡解。假设Robin函数有两个不同的指定临界点，每个临界点满足局部度条件。对每个足够小的ε>0，该系统存在非负弱解，其两个分量均非平凡，每个分量具有单气泡轮廓并集中在一个指定点，且每个分量在以另一个集中点为中心的球内恒为零；经自然气泡长度缩放后，该球的半径趋于无穷。主要障碍在于p=N/(N-2)∈(1,2)，因此当一个分量消失而另一个非零时，相互作用势(s,t)↦|s|^p|t|^p的梯度不可微，无法直接应用通常的全局Lyapunov--Schmidt约化。我们先通过变分法求解非线性外部问题，得到的死核消除了内部局域区域内的跨分量耦合，随后可在该区域进行投影临界约化，剩余的尺度和中心方程通过Brouwer度理论求解。
英文摘要
We construct segregated bubbling solutions for a two-component critical Schrödinger system of Brezis--Nirenberg type in a smooth bounded domain $Ω\subset\mathbb R^N$, $N\geq5$, 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 $p=N/(N-2)\in(1,2)$, so the gradient of the interaction potential $(s,t)\mapsto |s|^p|t|^p$ 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.
