分数阶对数薛定谔方程基态的定量分析

中文摘要 AI 辅助

设 $N\geq1$ 且 $0<s<1$。我们研究分数阶对数薛定谔方程 \begin{equation*} (-\Delta)^sQ=Q\log Q \quad\text{in }\mathbb{R}^N \end{equation*} 的正基态。我们证明对于每个 $N\geq1$ 和 $0<s<1$，正基态在平移意义下是唯一的且非简并的。更精确地，对于线性化算子 $L_Q=(-\Delta)^s-1-\log Q$，有 \begin{equation*} \ker L_Q=\operatorname{span}\{\partial_{x_1}Q,\cdots,\partial_{x_N}Q\}。 \end{equation*} 一个主要困难是势函数 $-1-\log Q$ 在 $\mathbb{R}^N$ 中无界，这阻碍了直接应用现有的针对有界势的分数阶薛定谔算子的径向振荡理论。我们通过有界势逼近来克服这一困难。同时利用相关的二次型具有 Morse 指标一以及角分解，我们得到了非简并性。基于对数基态的隔离性和分数幂方程的唯一定理，我们通过具有次临界幂非线性的变分逼近证明了唯一性。作为应用，我们建立了尖锐的分数阶对数 Sobolev 不等式并刻画了所有等号成立的情形。

英文摘要

Let $N\geq1$ and $0<s<1$. We study positive ground states of the fractional logarithmic Schrödinger equation \begin{equation*} (-Δ)^sQ=Q\log Q \quad\text{in }\mathbb{R}^N. \end{equation*} We prove that for every $N\geq1$ and $0<s<1$, the positive ground state is unique up to translations and nondegenerate. More precisely, for the linearized operator $L_Q=(-Δ)^s-1-\log Q$, it holds that \begin{equation*} \ker L_Q=\operatorname{span}\{\partial_{x_1}Q,\cdots,\partial_{x_N}Q\}. \end{equation*} A main difficulty is that the potential $-1-\log Q$ is unbounded in $\mathbb{R}^N$, which prevents a direct application of the available radial oscillation theory for fractional Schrödinger operators with bounded potentials. We overcome this difficulty by a bounded-potential approximation. Using also the fact that the associated quadratic form has Morse index one and an angular decomposition, we obtain the nondegeneracy. Based on the isolation of logarithmic ground states and the uniqueness theory for the fractional power equation, we prove uniqueness by a variational approximation with subcritical power nonlinearities. As an application, we establish sharp fractional logarithmic Sobolev inequalities and characterize all cases of equality.

发表机构

• Guizhou University of Finance and Economics（贵州财经大学）
• Central China Normal University（华中师范大学）

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