AI 中文总结
针对N≥5的“零质量”拟线性薛定谔方程，研究两类正径向解集的紧性，证明最小能量正径向解集非空且紧，严格次临界情形下有限L²质量正径向解集也在L²空间中紧，证明依赖变换后方程解的一致衰减估计。
AI 中文摘要
我们研究“零质量”拟线性薛定谔方程 \\( -\Delta u-u\Delta(|u|^2)=g(u) \quad\text{在}\mathbb R^N\\)（\\(N\ge5\\)）的两类正径向解集的紧性。对于严格次临界或在无穷远处渐近临界的非线性项，我们证明最小能量正径向解的集合非空且在自然空间中紧。此外，在严格次临界情形及额外假设下，我们表明所有有限L²质量的正径向解的集合在\\(L^2(\mathbb R^N)\\)中紧。该证明依赖于变换后的半线性方程对应正径向解的一致衰减估计，其给出了所需的一致L²尾控制。
英文摘要
We study the compactness of two sets of positive radial solutions to ``zero mass'' quasi-linear Schrödinger equation \[ -Δu-uΔ(|u|^2)=g(u) \quad\text{in }\mathbb R^N,\quad N\ge5. \] For nonlinearities that are either strictly subcritical or asymptotically critical at infinity, we prove that the set of least energy positive radial solutions is nonempty and compact in the natural space. Moreover, in the strictly subcritical case and under additional assumptions, we show that the set of all finite $L^2$-mass positive radial solutions is compact in \(L^2(\mathbb R^N)\). The proof relies on uniform decay estimates for the corresponding positive radial solutions of the transformed semilinear equation, which yield the required uniform \(L^2\)-tail control.
