大质量伪相对论非线性薛定谔方程正径向基态的唯一性
Uniqueness of positive radial ground states for the massive pseudo-relativistic nonlinear Schrödinger equation
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中文总结 AI 辅助
研究大质量伪相对论非线性薛定谔方程正径向基态的唯一性，通过质量协变振荡定理和延拓论证，证明其唯一性并修复尺度不变性失效问题。
中文摘要 AI 辅助
我们研究半线性伪相对论方程 \\[ \sqrt{-\Delta+m^{2}}\\,Q+\omega Q=Q^{p-1}\qquad\text{在 }\R^{d} \\] 其中 $m>0$，$\omega>0$，$d\ge1$ 且 $2<p<2^{*}:=\frac{2d}{d-1}$（当 $d=1$ 时 $2^{*}=\infty$）。我们证明正径向变分基态是唯一的。证明依赖于线性化算子的质量协变振荡定理，该定理通过纯变分约化建立：将 $\sqrt{-\Delta+m^{2}}$ 的 Caffarelli--Silvestre--Duffin 延拓与基态代入相结合，将线性化特征值问题约化为无质量、无势能的加权 Steklov 问题，其第二特征函数通过加权 Dirichlet 能量在符号分解下的严格超可加性恰好改变一次符号。线性化的非退化性则由两个正交关系获得；对于 $m>0$，尺度不变性的失效破坏了无质量 Frank--Lenzmann--Silvestre 理论中使用的第二个关系，通过质量协变恒等式加以修复。全局唯一性通过以 $m=0$ 为锚点（由 Frank--Lenzmann--Silvestre 定理保证）的质量参数延拓论证得出。
英文摘要
We study the semilinear pseudo-relativistic equation \[ \sqrt{-Δ+m^{2}}\,Q+ωQ=Q^{p-1}\qquad\text{in }\R^{d}, \] with $m>0$, $ω>0$, $d\ge1$ and $2<p<2^{*}:=\frac{2d}{d-1}$ (with $2^{*}=\infty$ when $d=1$). We prove that the positive radial variational ground state is unique. The proof rests on a mass-covariant oscillation theorem for the linearized operator, established through a purely variational reduction: the Caffarelli--Silvestre--Duffin extension for $\sqrt{-Δ+m^{2}}$ combined with a ground-state substitution reduces the linearized eigenvalue problem to a massless, potential-free weighted Steklov problem, whose second eigenfunction changes sign exactly once by strict superadditivity of the weighted Dirichlet energy under sign decomposition. The nondegeneracy of the linearization is then obtained from two orthogonality relations; the failure of scale invariance for $m>0$, which destroys the second relation used in the massless Frank--Lenzmann--Silvestre theory, is repaired by a mass-covariance identity. Global uniqueness follows by a continuation argument in the mass parameter, anchored at $m=0$ by the theorem of Frank--Lenzmann--Silvestre.
发表机构
- University of Science and Technology Beijing（北京科技大学）
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