CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS

doi:10.1007/s10473-025-0511-1

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 1981-2009.doi: 10.1007/s10473-025-0511-1

CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS

Lintao LIU¹, Shuai YAO², Kaimin TENG³, Haibo CHEN^4,*

1. Department of Mathematics, North University of China, Taiyuan 030051, China;
2. School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China;
3. Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China; School of Mathematics and Statistics, Central South University, Changsha 410083, China

收稿日期:2023-07-31 修回日期:2024-11-19 出版日期:2025-09-25 发布日期:2025-10-14

CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS

Lintao LIU¹, Shuai YAO², Kaimin TENG³, Haibo CHEN^4,*

1. Department of Mathematics, North University of China, Taiyuan 030051, China;
2. School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China;
3. Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China; School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received:2023-07-31 Revised:2024-11-19 Online:2025-09-25 Published:2025-10-14
Contact: ^*Haibo Chen, E-mail: math_chb@csu.edu.cn
About author:Lintao Liu, E-mail: liulintao1995@163.com; Shuai Yao, E-mail: shyao2019@163.com; Kaimin Teng, E-mail: tengkaimin2013@163.com
Supported by:
Liu's research was supported by the Fundamental Research Program of Shanxi Province (202403021222126). Teng's research was supported by the Fundamental Research Program of Shanxi Province (202303021211056). Chen's research was supported by the National Natural Science Foundation of China (12071486).

摘要/Abstract

摘要： We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential. By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when $a\nearrow a^{\ast}:=\|Q\|_{2}^{2s}$, where $Q$ is the unique positive radial solution of $(-\Delta)^{s}u+su-|u|^{2s}u=0$ in $\mathbb{R}^{2}$. Based on the discussion of the concentration phenomenon, we prove the local uniqueness of minimizers by establishing a local Pohožaev identity and studying the blow-up estimates to the nonlocal operator $(-\Delta)^{s}$.

关键词: concentration, energy estimates, asymptotic uniqueness, Pohožaev identity

Abstract: We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential. By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when $a\nearrow a^{\ast}:=\|Q\|_{2}^{2s}$, where $Q$ is the unique positive radial solution of $(-\Delta)^{s}u+su-|u|^{2s}u=0$ in $\mathbb{R}^{2}$. Based on the discussion of the concentration phenomenon, we prove the local uniqueness of minimizers by establishing a local Pohožaev identity and studying the blow-up estimates to the nonlocal operator $(-\Delta)^{s}$.

Key words: concentration, energy estimates, asymptotic uniqueness, Pohožaev identity

中图分类号:

35J50

Lintao LIU, Shuai YAO, Kaimin TENG, Haibo CHEN. CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS[J]. 数学物理学报(英文版), 2025, 45(5): 1981-2009.

Lintao LIU, Shuai YAO, Kaimin TENG, Haibo CHEN. CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS[J]. Acta mathematica scientia,Series B, 2025, 45(5): 1981-2009.

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CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS

CONCENTRATION AND UNIQUENESS OF MINIMIZERS FOR FRACTIONAL SCHRÖDINGER ENERGY FUNCTIONALS

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