ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

doi:10.1007/s10473-025-0403-4

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1265-1283.doi: 10.1007/s10473-025-0403-4

ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

Jun WANG, Xiaoguang LI^*

School of Mathematical Science and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, China

收稿日期:2023-09-22 出版日期:2025-10-10 发布日期:2025-10-10

ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

Jun WANG, Xiaoguang LI^*

School of Mathematical Science and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, China

Received:2023-09-22 Online:2025-10-10 Published:2025-10-10
Contact: ^*Xiaoguang LI, E-mail: lixgmath@163.com
About author:Jun WANG, E-mail: wangjunmath1996@163.com
Supported by:
National Natural Science Foundation of China (11771314, 12071323).

摘要/Abstract

摘要： We study the following minimization problem: $$d_{p}(M_{p}):=\inf\{E_{p}(u): \|u\|_{L^{2}}=M_{p}\},$$ where the Gross-Pitaevskii energy functional $$E_{p}(u)=\int_{\mathbb{R}^{N}}|\nabla u|^{2}-c\frac{|u|^{2}}{|x|^{2}}+V(x)|u|^{2}{\rm d}x-\frac{2}{p+2}\int_{\mathbb{R}^{N}}|u|^{p+2}{\rm d}x.$$ When $p=p^{*}:=\frac{4}{N}$, the precise concentration behavior of minimizers is analyzed as $M_{p^{*}}\nearrow \|Q_{p^{*}}\|_{L^{2}}$, where $Q_{p^{*}}$ is the unique radially positive solution of $-\Delta \varphi-c\frac{\varphi}{|x|^{2}}-|\varphi|^{p^{*}+1}\varphi=0$. When $0<p<p^{*}$, we prove that all minimizers must blow up if $\lim\limits_{p\to p^{*}}M_{p}\geq \|Q_{p^{*}}\|_{L^{2}}$. On this argument, the detailed concentration behavior of minimizers is established as $p\nearrow p^{*}$.

关键词: minimization problem, minimizers, concentration behavior

Abstract: We study the following minimization problem: $$d_{p}(M_{p}):=\inf\{E_{p}(u): \|u\|_{L^{2}}=M_{p}\},$$ where the Gross-Pitaevskii energy functional $$E_{p}(u)=\int_{\mathbb{R}^{N}}|\nabla u|^{2}-c\frac{|u|^{2}}{|x|^{2}}+V(x)|u|^{2}{\rm d}x-\frac{2}{p+2}\int_{\mathbb{R}^{N}}|u|^{p+2}{\rm d}x.$$ When $p=p^{*}:=\frac{4}{N}$, the precise concentration behavior of minimizers is analyzed as $M_{p^{*}}\nearrow \|Q_{p^{*}}\|_{L^{2}}$, where $Q_{p^{*}}$ is the unique radially positive solution of $-\Delta \varphi-c\frac{\varphi}{|x|^{2}}-|\varphi|^{p^{*}+1}\varphi=0$. When $0<p<p^{*}$, we prove that all minimizers must blow up if $\lim\limits_{p\to p^{*}}M_{p}\geq \|Q_{p^{*}}\|_{L^{2}}$. On this argument, the detailed concentration behavior of minimizers is established as $p\nearrow p^{*}$.

Key words: minimization problem, minimizers, concentration behavior

Jun WANG, Xiaoguang LI. ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL[J]. 数学物理学报(英文版), 2025, 45(4): 1265-1283.

Jun WANG, Xiaoguang LI. ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1265-1283.

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ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

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