Hostname: page-component-7b9c58cd5d-6tpvb Total loading time: 0 Render date: 2025-03-15T14:10:54.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Concentration phenomena for fractional magnetic NLS equations

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  28 May 2021

Vincenzo Ambrosio Open the ORCID record for Vincenzo Ambrosio [Opens in a new window]
Vincenzo Ambrosio*
Affiliation:
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131, Ancona, Italy (v.ambrosio@staff.univpm.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Keywords

Fractional magnetic operatorsLjusternik–Schnirelman theoryvariational methods

MSC classification

Primary: 35A15: Variational methods
Secondary: 35R11: Fractional partial differential equations 35S05: Pseudodifferential operators 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 479 - 517
Check for updates
Copyright
Copyright © The Author(s), 2021. 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

Alves, C. O. and Figueiredo, G. M.. Multiple solutions for a semilinear elliptic equation with critical growth and magnetic field. Milan J. Math. 82 (2014), 389–405.CrossRefGoogle Scholar
Alves, C. O., Figueiredo, G. M. and Furtado, M. F.. Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 36 (2011), 1565–1586.CrossRefGoogle Scholar
Alves, C. O. and Miyagaki, O. H.. Existence and concentration of solution for a class of fractional elliptic equation in ${\mathbb {R}^{N}}$ via penalization method. Calc. Var. Partial Differ. Equ. 55 (2016), 19pp. art. 47.CrossRefGoogle Scholar
Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
Ambrosio, V.. Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method. Ann. Mat. Pura Appl. (4) 196 (2017), 2043–2062.CrossRefGoogle Scholar
Ambrosio, V.. Concentrating solutions for a class of nonlinear fractional Schrödinger equations in ${\mathbb {R}^{N}}$. Rev. Mat. Iberoam. 35 (2019), 1367–1414.CrossRefGoogle Scholar
Ambrosio, V.. Existence and concentration results for some fractional Schrödinger equations in ${\mathbb {R}^{N}}$ with magnetic fields. Commun. Partial Differ. Equ. 44 (2019), 637–680.CrossRefGoogle Scholar
Ambrosio, V.. On a fractional magnetic Schrödinger equation in $\mathbb {R}$ with exponential critical growth. Nonlinear Anal. 183 (2019), 117–148.CrossRefGoogle Scholar
Ambrosio, V.. Concentration phenomena for a fractional Choquard equation with magnetic field. Dyn. Partial Differ. Equ. 16 (2019), 125–149.CrossRefGoogle Scholar
Ambrosio, V.. Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Dis. Contin. Dyn. Syst. 40 (2020), 781–815.CrossRefGoogle Scholar
Ambrosio, V.. Existence and concentration of nontrivial solutions for a fractional magnetic Schrödinger–Poisson type equation. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) XXI (2020), 1023–1061.Google Scholar
Ambrosio, V.. A local mountain pass approach for a class of fractional NLS equations with magnetic fields. Nonlinear Anal. 190 (2020), 14 pp. 111622.CrossRefGoogle Scholar
Ambrosio, V. and d'Avenia, P.. Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity. J. Diff. Equ. 264 (2018), 3336–3368.CrossRefGoogle Scholar
Arioli, G. and Szulkin, A.. A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170 (2003), 277–295.CrossRefGoogle Scholar
Avron, J., Herbst, I. and Simon, B.. Schrödinger operators with magnetic fields I. General interactions. Duke Math. J. 45 (1978), 847–883.CrossRefGoogle Scholar
Benci, V. and Cerami, G.. Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 2 (1994), 29–48.CrossRefGoogle Scholar
Bucur, C. and Valdinoci, E.. Nonlocal diffusion and applications. Lecture Notes of the Unione Matematica Italiana, vol. 20 (Bologna: Springer, [Cham]; Unione Matematica Italiana, 2016, xii+155 pp.).Google Scholar
Chabrowski, J. and Szulkin, A.. On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Methods Nonlinear Anal. 25 (2005), 3–21.CrossRefGoogle Scholar
Cingolani, S. and Secchi, S.. Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275 (2002), 108–130.CrossRefGoogle Scholar
Dávila, J., del Pino, M. and Wei, J.. Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Diff. Equ. 256 (2014), 858–892.CrossRefGoogle Scholar
d'Avenia, P. and Squassina, M.. Ground states for fractional magnetic operators. ESAIM Control Optim. Calc. Var. 24 (2018), 1–24.CrossRefGoogle Scholar
Del Pezzo, L. M. and Quaas, A.. A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian. J. Diff. Equ. 263 (2017), 765–778.CrossRefGoogle Scholar
Del Pezzo, L. M. and Quaas, A.. Spectrum of the fractional $p$-Laplacian in ${\mathbb {R}^{N}}$ and decay estimate for positive solutions of a Schrödinger equation. Nonlinear Anal. 193 (2020), 111479.CrossRefGoogle Scholar
del Pino, M. and Felmer, P. L.. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996), 121–137.CrossRefGoogle 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
Dipierro, S., Medina, M. and Valdinoci, E.. Fractional elliptic problems with critical growth in the whole of ${\mathbb {R}^{n}}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] vol. 15, viii+152 pp. (Pisa: Edizioni della Normale, 2017).Google Scholar
Ekeland, I.. On the variational principle. J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
Esteban, M. and Lions, P. L.. Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In F. Colombini, A. Marino, L. Modica and S. Spagnolo (eds), Partial differential equations and the calculus of variations, Vol. I, Progr. Nonlinear Differential Equations Appl., vol. 1, pp. 401–449 (Boston, MA: Birkhäuser Boston, 1989).Google Scholar
Fall, M. M., Mahmoudi, F. and Valdinoci, E.. Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28 (2015), 1937–1961.CrossRefGoogle Scholar
Felmer, P., Quaas, A. and Tan, J.. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.CrossRefGoogle Scholar
Figueiredo, G. M. and Santos, J. R.. Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method. ESAIM Control Optim. Calc. Var. 20 (2014), 389–415.CrossRefGoogle Scholar
Figueiredo, G. M. and Siciliano, G.. A multiplicity result via Ljusternick–Schnirelman category and Morse theory for a fractional Schrödinger equation in $\mathbb {R}^{N}$. NoDEA Nonlinear Diff. Equ. Appl. 23 (2016), 22 pp., Art. 12.Google Scholar
Fiscella, A., Pinamonti, A. and Vecchi, E.. Multiplicity results for magnetic fractional problems. J. Diff. Equ. 263 (2017), 4617–4633.CrossRefGoogle Scholar
Hiroshima, F., Ichinose, T. and Lörinczi, J.. Kato's inequality for magnetic relativistic Schrödinger operators. Publ. Res. Inst. Math. Sci. 53 (2017), 79–117.CrossRefGoogle Scholar
Ichinose, T.. Magnetic relativistic Schrödinger operators and imaginary-time path integrals. In M. Demuth and W. Kirsch (eds), Mathematical physics, spectral theory and stochastic analysis. Oper. Theory Adv. Appl., vol. 232, pp. 247–297 (Basel: Birkhäuser/Springer Basel AG, 2013).Google Scholar
Kato, T.. Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13 (1973), 135–148.CrossRefGoogle Scholar
Kurata, K.. Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 41 (2000), 763–778.CrossRefGoogle Scholar
Laskin, N.. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000), 298–305.CrossRefGoogle Scholar
Mingqi, X., Rădulescu, V. D. and Zhang, B.. A critical fractional Choquard–Kirchhoff problem with magnetic field. Commun. Contemp. Math. 21 (2019), 36 pp., 1850004.CrossRefGoogle Scholar
Moser, J.. A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13 (1960), 457–468.CrossRefGoogle Scholar
Nguyen, H. M., Pinamonti, A., Squassina, M. and Vecchi, E.. New characterizations of magnetic Sobolev spaces. Adv. Nonlinear Anal. 7 (2018), 227–245.CrossRefGoogle Scholar
Palatucci, G. and Pisante, A.. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50 (2014), 799–829.CrossRefGoogle Scholar
Rabinowitz, P. H.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270–291.CrossRefGoogle Scholar
Silvestre, L.. Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math., 60 (2007), 67–112.CrossRefGoogle Scholar
Squassina, M. and Volzone, B.. Bourgain–Brezis–Mironescu formula for magnetic operators. C. R. Math. 354 (2016), 825–831.CrossRefGoogle Scholar
Sulem, C. and Sulem, P.-L.. The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, vol. 139, xvi+350 pp. (New York: Springer-Verlag, 1999).Google Scholar
Szulkin, A. and Weth, T.. The method of Nehari manifold. In David Yang Gao and D. Motreanu (eds), Handbook of Nonconvex Analysis and Applications, pp. 597–632 (Somerville, MA: Int. Press, 2010).Google Scholar
Willem, M.. Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24 (Boston, MA: Birkhäuser Boston, Inc., 1996).Google Scholar