R^n中奇异Q-曲率方程的唯一性与对称破缺
Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$
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中文总结 AI 辅助
该研究针对R^n中奇异Q-曲率方程，通过变换建立解的对称性，证明径向解的唯一性与非退化性，构造非径向解分支，还证明γ足够小时所有解均为径向解。
中文摘要 AI 辅助
对于n≥3且γ>-1，我们研究方程(-Δ)^{n/2}u=(n-1)!|x|^{nγ}e^{nu}在R^n中的正规解。通过Emden–Fowler变换，我们将该问题重新表述到柱面上。首先，我们建立正规解的Kelvin对称性；接着，证明径向正规解在相差一个伸缩变换下的存在性与唯一性，并确立其径向非退化性；随后，对线性化算子进行详细谱分析，利用Crandall–Rabinowitz定理构造非径向正规解的分支；最后，证明当γ>0足够小时，所有正规解均为径向解。
英文摘要
For $n\ge3$ and $γ>-1$, we study normal solutions of \begin{equation*} (-Δ)^{n/2}u=(n-1)!\,|x|^{nγ}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when $γ>0$ is sufficiently small.
