发表机构
School of Mathematics, Jilin University; School of Mathematics & International Institute for Mathematical Sciences, Jilin University（吉林大学数学学院; 吉林大学数学学院与国际数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究对数非线性伪相对论薛定谔方程的基态，通过幂律近似建立存在性与定性性质，证明非相对论极限收敛于高斯子，并推广到能量基态。
AI 中文摘要
本文研究具有对数非线性的伪相对论薛定谔方程 (√(-c²Δ+m²c⁴)-mc²)u+μu=u log|u| 在 R^N 中。通过幂律近似，建立了作用基态的存在性和定性性质。然后证明当 c→∞ 时，这些态在 H¹(R^N) 中收敛到高斯子。对于足够大的 c，进一步获得了作用基态在平移和符号变化下的唯一性以及非简并性。最后，证明了作用基态与能量基态之间的精确对应关系，从而将前述结果推广到任意给定质量下的能量基态。
英文摘要
In this paper, we study the pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \begin{equation*} \left(\sqrt{-c^{2}Δ+m^{2}c^{4}}-mc^{2}\right)u+μu=u\log|u| \quad\text{in }\mathbb{R}^{N}. \end{equation*} The existence and qualitative properties of action ground states are established, by means of a power-law approximation. We then prove that these states converge to the Gausson in $H^{1}(\mathbb{R}^{N})$ as $c\to\infty$. For sufficiently large $c$, we further obtain uniqueness of action ground states up to translation and sign change, as well as their nondegeneracy. Finally, we prove an exact correspondence between action and energy ground states, thereby extending the preceding results to energy ground states at arbitrary prescribed mass.
