Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T09:26:30.672Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cont, R. and Tankov, P.. Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series (Boca Raton: Chapman & Hall/CRC, 2004).Google Scholar
Chang, S. Y. A. and del Mar González, M.. Fractional Laplacian in conformal geometry. Adv. Math. 226 (2011), 1410–1432.CrossRefGoogle Scholar
Cheng, M.. Bound state for the fractional Schródinger equation with unbounded potential. J. Math. Phys. 53 (2012), 043507–043507–7.CrossRefGoogle Scholar
Cabré, X. and Sire, Y.. Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.CrossRefGoogle Scholar
Deren, F. Y., Cerdik, T. S. and Agarwal, R. P.. Existence criteria of positive solutions for fractional p-Laplacian boundary value problems. Filomat 34 (2020), 3789–3799.Google Scholar
L.X. Tian, M. D, Wang, J. and Zhang, F. B.. Existence of normalized solutions for nonlinear fractional Schródinger equations with trapping potentials. Proc. Roy. Soc. Edinburgh 149 (2019), 617–653.Google Scholar
Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.CrossRefGoogle Scholar
Felmer, P., Quaas, A. and Tan, J. G.. Positive solutions of the nonlinear Schródinger equation with the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.CrossRefGoogle Scholar
Guo, Y. J. and Seiringer, R.. On the mass concentration for Bose-Einstein condensation with attractive interactions. Lett. Math. Phys. 104 (2013), 141–156.CrossRefGoogle Scholar
Guo, Y. J., Wang, Z.-Q., Zeng, X. Y. and Zhou, H.-S.. Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials. Nonlinearity 31 (2018), 957–979.CrossRefGoogle Scholar
Guo, Y. J., Zeng, X. Y. and Zhou, H.-S.. Energy estimates and symmetry breaking in attractive Bose-Einstein condensation with ring-shaped potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 809–828.Google Scholar
Laskin, N.. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000), 298–305.CrossRefGoogle Scholar
Laskin, N.. Fractional Schródinger equation. Phys. Rev. 66 (2002), 56–108.Google ScholarPubMed
Liu, Z. S. and Ouyang, Z. G.. Existence of positive ground state solutions for fractional Schródinger equations with a general nonlinearity. Appl. Anal. 97 (2018), 1154–1171.CrossRefGoogle Scholar
Metzler, R. and Klafter, J.. The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.CrossRefGoogle Scholar
Ni, W. M. and Takagi, I.. On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 44 (1991), 819–851.CrossRefGoogle Scholar
Ragusa, M. A.. Parabolic Herz spaces and their applications. Appl. Math. Lett. 25 (2012), 1270–1273.CrossRefGoogle Scholar
Silvestre, L.. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), 67–112.CrossRefGoogle Scholar
Secchi, S.. Ground state solutions for nonlinear fractional Schródinger equations in $\mathbb {R}^N$. J. Math. Phys. 54 (2013), 03501.CrossRefGoogle Scholar
Su, Y., Chen, H. B., Liu, S. L. and Fang, X. W.. Fractional Schródinger-Poisson systems weighted Hardy potential and critical exponent. Electron. J. Differ. Equ. 2020 (2020), 1–17.Google Scholar
Teng, K. M. and Agarwal, R. P.. Existence and concentration of positive ground state solutions for nonlinear fractional Schródinger-Poisson system with critical growth. Math. Methods Appl. Sci. 42 (2018), 8258–8293.CrossRefGoogle Scholar
Teng, K. M. and Cheng, Y. Q.. Multiplicity and concentration of nontrivial solutions for fractional Schródinger-Poisson system involving critical growth. Nonlinear Anal. 202 (2021), 112144.CrossRefGoogle Scholar
Teng, K. M. and Wu, X.. Concentration of bound states for fractional Schródinger-Poisson system via penalization methods. Commun. Pure Appl. Anal. 21 (2022), 1157–1187.CrossRefGoogle Scholar
Wu, Y. and Taarabti, S.. Existence of two positive solutions for two kinds of fractional p-Laplacian equations. J. Funct. Spaces 2021 (2021), 1–9.Google Scholar
Wang, X. F.. On concentration of positive bound-states of nonlinear Schródinger-equations. Comm. Math. Phys. 153 (1993), 229–244.CrossRefGoogle Scholar