Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T03:12:23.334Z Has data issue: false hasContentIssue false
  • English
  • Français

On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

Part of: Equations of mathematical physics and other areas of application Elliptic equations and systems

Published online by Cambridge University Press:  30 January 2019

Woocheol Choi ,
Younghun Hong  and
Jinmyoung Seok
Woocheol Choi
Affiliation:
Department of Mathematics Education, Incheon National University, Incheon22012, Republic of Korea ([email protected])
Younghun Hong
Affiliation:
Department of Mathematics, Chung-Ang University, Seoul06974, Republic of Korea ([email protected])
Jinmyoung Seok
Affiliation:
Department of Mathematics, Kyonggi University, Suwon16227, Republic of Korea ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation

$$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$
involving an H1/2-critical/supercritical power-type nonlinearity, that is, p ⩾ ((n + 1)/(n − 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H1/2-critical/supercritical but it is H1-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either (i) if the nonlinearity is H1/2-critical/supercritical in the ultra-relativistic regime or (ii) if the nonlinearity is H1-critical/supercritical in all cases.

Keywords

pseudo-relativistic NLSsupercriticalityexistencenon-existence

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 35J61: Semilinear elliptic equations 35Q51: Soliton-like equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1241 - 1263
Copyright
Copyright © 2019 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

1Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal., 82 (1983), 313–345.CrossRefGoogle Scholar
2Choi, W. and Seok, J.. Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 021510, 15 pp.CrossRefGoogle Scholar
3Choi, W., Hong, Y. and Seok, J.. Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations, arXiv:1610.06030.Google Scholar
4Cingolani, S. and Secchi, S.. Ground states for the pseudo-relativistic Hartree equation with external potential, in press on Proceedings of the Royal Society of Edinburgh.Google Scholar
5Cingolani, S. and Secchi, S.. Semiclassical analysis for pseudo-relativistic Hartree equations. J. Differential Equations, 258 (2015), 4156–4179.CrossRefGoogle Scholar
6Comech, A.. Guan, M., and Gustafson, S., On linear instability of solitary waves for the nonlinear Dirac equation, Ann. Inst. H. Poincar Anal. Non Linéaire, 31 (2014), 639–654.CrossRefGoogle Scholar
7Coti-Zelati, V. and Nolasco, M.. Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), 51–72.CrossRefGoogle Scholar
8Coti-Zelati, V. and Nolasco, M.. Ground states for pseudo-relativistic Hartree equations of critical type. Rev. Mat. Iberoam 29 (2013), 1421–1436.CrossRefGoogle Scholar
9Frank, R. L. and Lenzmann, E.. Uniqueness of non-linear ground states for fractional Laplacians in ℝ. Acta Math. 210 (2013), 261–318.CrossRefGoogle Scholar
10Frank, R. L., Lenzmann, E. and Silvestre, L.. Uniqueness of radial solutions for the fractional Laplacian. Comm. Pure Appl. Math. 69 (2016), 1671–1726.CrossRefGoogle Scholar
11Grafakos, L.. Classical fourier analysis, 2nd edn, Graduate Texts in Mathematics, 249 (New York: Springer, 2008).Google Scholar
12Guan, M.. Solitary wave solutions for the nonlinear Dirac equations, arXiv:0812.2273.Google Scholar
13Kwong, M. K.. Uniqueness of positive solutions of Δu − u + u p = 0 in ℝn. Arch. Rational Mech. Anal. 105 (1989), 243–266.CrossRefGoogle Scholar
14Micheletti, A. M., Musso, M. and Pistoia, A.. Super-position of spikes for a slightly supercritical elliptic equation in open R N. Discrete Contin. Dyn. Syst. 12 (2005), 747–760.CrossRefGoogle Scholar
15Mugnai, D.. Pseudorelativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud. 13 (2013), 799–823.CrossRefGoogle Scholar
16Ounaies, H.. Perturbation method for a class of nonlinear Dirac equations. Diff. Int. Equ. 13 (2000), 707–720.Google Scholar
17Strauss, W.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
18Tan, J., Wang, Y. and Yang, J.. Nonlinear fractional field equations. Nonlinear Anal. 75 (2012), 2098–2110.CrossRefGoogle Scholar