含自聚焦核的薛定谔方程解的存在性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 1721-1741.

含自聚焦核的薛定谔方程解的存在性

刘浚源¹(), 魏欣然²^,^*()

¹ 华中农业大学信息学院武汉 430070
² 信息工程大学郑州 450001

收稿日期:2025-03-31 修回日期:2026-01-19 出版日期:2026-10-26 发布日期:2026-09-07
通讯作者: 魏欣然 E-mail:jyliu@mail.hzau.edu.cn;weichenxi444@163.com
作者简介:刘浚源, E-mail:jyliu@mail.hzau.edu.cn
基金资助:
中央高校基本科研业务费专项资金(2662025XXQD003);信息工程大学校自主科研项目

Sign-Changing Solutions for Schrödinger Equations with Shrinking Self-Focusing Core

Junyuan Liu¹(), Xinran Wei²^,^*()

¹ College of Informatics, Huazhong Agricultural University, Wuhan 430070
² Information Engineering University, Zhengzhou 450001

Received:2025-03-31 Revised:2026-01-19 Online:2026-10-26 Published:2026-09-07
Contact: Xinran Wei E-mail:jyliu@mail.hzau.edu.cn;weichenxi444@163.com
Supported by:
Fundamental Research Funds for the Central Universities(2662025XXQD003);Independent Scientific Rescarch Project of Information Engineering University

摘要/Abstract

摘要：

该文主要探究 Schrödinger 方程 $- \Delta u + u = {Q_n}(x){\left| u \right|^{p - 2}}u, x \in \mathbb{R}^N$ 的变号解, 其中 $Q_n$ 是一个具体的有界函数, 它有一个自聚焦核, 即当 $n\rightarrow \infty$ 时, ${{\rm supp}\{Q_n^+>0\}}$ 收敛到一个有限点集. 最终证明了若自聚焦核只收敛到一个或两个点, 则对任意 $n>0$, 该方程存在仅变号一次的最小能量变号解. 并且, 该最小能量变号解也会收敛到某个极限方程. 该文的创新点在于使用了不同于 Fang 和 Wang [Fang X D, Wang Z Q. Calc Var, 2020, 59(4): Art 129] 中的一种全新的罚函数构造了集中形式的局部有界变号解.

关键词: 变号解, 薛定谔方程, 存在性

Abstract:

We are interested in the sign-changing solutions for the equation $- \Delta u + u = {Q_n}(x){\left| u \right|^{p - 2}}u$ in $\mathbb{R}^N$, where $Q_n$ are concrete bounded functions with a self-focusing core ${{\rm supp}\{Q_n^+>0\}}$ that shrinks to a finite set of points as $n\rightarrow \infty$. We prove the existence of least energy sign-changing solutions that change sign only once for any $n>0$ if the self-focusing core shrinks to one or two points. Additionally, the least energy sign-changing solutions may also concentrate and converge to the solution of some limit equation. Furthermore, we utilize a penalty function distinct from that in Fang and Wang [Fang X D, Wang Z Q. Calc Var, 2020, 59(4): Art 129] to construct localized, bounded, sign-changing solutions of concentration type.

Key words: sign-changing solutions, Schrödinger equations, existence

中图分类号:

O175.29

刘浚源, 魏欣然. 含自聚焦核的薛定谔方程解的存在性[J]. 数学物理学报, 2026, 46(5): 1721-1741.

Junyuan Liu, Xinran Wei. Sign-Changing Solutions for Schrödinger Equations with Shrinking Self-Focusing Core[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1721-1741.

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