AI 中文总结
该数学研究确定了尖锐索伯列夫不等式标准径向极值处线性化p-拉普拉斯算子的完整特征基，通过球谐分解等方法实现，索引含角度数与径向模式数。
AI 中文摘要
设1<p<n，且v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}为尖锐索伯列夫不等式的标准径向极值。我们确定在v处线性化p-拉普拉斯算子的所有特征值与特征空间，该算子由L²(ℝⁿ, v^{p^*-2}dx)中的闭二次型定义。经球谐函数分解后，显式规范变换与变量替换将每个径向算子识别为平移雅可比算子，由此得到由(ℓ,k)∈ℕ₀²索引的完整特征基，其中ℓ为角度数，k为径向模式数。
英文摘要
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.
