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Concentration behaviour of normalized ground states of the mass critical fractional Schrödinger equations with ring-shaped potentials

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  14 December 2022

Lintao Liu ,
Kaimin Teng ,
Jie Yang  and
Haibo Chen
Lintao Liu
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China (liulintao1995@163.com)
Kaimin Teng
Affiliation:
Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, PR China (tengkaimin2013@163.com)
Jie Yang
Affiliation:
School of Mathematics and Computational Science, Huaihua University, Huaihua, Hunan 418008, PR China (dafeyang@163.com; math_chb@163.com)
Haibo Chen
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China (liulintao1995@163.com)
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Abstract

We consider $L^{2}$-constraint minimizers of the mass critical fractional Schrödinger energy functional with a ring-shaped potential $V(x)=(|x|-M)^{2}$, where $M>0$ and $x\in \mathbb {R}^{2}$. By analysing some new estimates on the least energy of the mass critical fractional Schrödinger energy functional, we obtain the concentration behaviour of each minimizer of the mass critical 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 $\mathbb {R}^{2}$.

Keywords

Fractional Schrödinger equationsNormalized ground statesRing-shaped potentials

MSC classification

Secondary: 35J60: Nonlinear elliptic equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1993 - 2024
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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