高阶对数薛定谔方程非负束缚态的存在性

中文摘要 AI 辅助

本文研究高阶对数薛定谔方程 $$ -\alpha\Delta u+V(x)u-\beta u\ln|u| = \gamma u(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ 的非负束缚态的存在性，其中 $\alpha,\beta,\gamma>0$，$m$ 为奇数正整数，$V$ 为在无穷远处收敛到常数的正有界势。一阶与高阶对数项的同时出现导致了非光滑变分结构及额外的紧性困难。我们引入一族幂律逼近，并推导出当逼近指数趋于对数极限时一致的估计。在自治非线性项满足结构条件下，我们得到了具有连通正集的非负自治轮廓在平移意义下的唯一性，以及低于双轮廓阈值的相应能量间隙。小振幅行为表现出极大值原理/紧支撑二分性：$m=1$ 情形给出正性，而更高的奇数阶则落入自治轮廓的紧支撑区域。随后，我们建立了约束Palais-Smale序列的轮廓分解，并构造了基于重心的、低于分裂阈值的极小极大水平。这防止了通过多轮廓的质量损失，并使我们能够过渡到对数极限。在所述结构与能量条件下，我们得到了原方程的一个非平凡非负束缚态。

英文摘要

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -αΔu+V(x)u-βu\ln|u| = γu(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $α,β,γ>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a constant at infinity. The simultaneous presence of the first- and higher-order logarithmic terms leads to a nonsmooth variational structure and additional compactness difficulties. We introduce a family of power-law approximations and derive estimates that are uniform as the approximation exponent tends to the logarithmic limit. Under a structural condition on the autonomous nonlinearity, we obtain uniqueness, up to translations, of the nonnegative autonomous profile with connected positivity set and a corresponding energy gap below the two-profile threshold. The small-amplitude behavior exhibits a maximum-principle/compact-support dichotomy: the case $m=1$ yields positivity, whereas the higher odd orders fall into the compact-support regime for the autonomous profile. We then establish a profile decomposition for constrained Palais-Smale sequences and construct a barycenter-based min-max level below the splitting threshold. This prevents loss of mass through multiple profiles and allows us to pass to the logarithmic limit. Under the stated structural and energy conditions, we obtain a nontrivial nonnegative bound state of the original equation.

发表机构

• Faculty of Science, Kunming University of Science and Technology（昆明理工大学理学院）
• Research Center for Mathematics and Interdisciplinary Sciences, Kunming University of Science and Technology（昆明理工大学数学与交叉科学研究中心）
• City College, Kunming University of Science and Technology（昆明理工大学城市学院）

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