May  2022, 21(5): 1649-1672. doi: 10.3934/cpaa.2022038

Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials

1. 

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China

2. 

School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

* Corresponding author

Received  September 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: This work was partially supported by NSFC grants (No.12101150; No.11931012) and the Science and Technology Foundation of Guizhou Province ([2021]ZK008)

This paper deals with the following fractional magnetic Schrödinger equations
$ \varepsilon^{2s}(-\Delta)^s_{A/\varepsilon} u +V(x)u = |u|^{p-2}u, \ x\in{\mathbb R}^N, $
where
$ \varepsilon>0 $
is a parameter,
$ s\in(0,1) $
,
$ N\geq3 $
,
$ 2+2s/(N-2s)<p<2_s^*: = 2N/(N-2s) $
,
$ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $
with
$ \alpha\in(0,1] $
is a magnetic field,
$ V:{\mathbb R}^N\to{\mathbb R} $
is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of
$ V $
as
$ \varepsilon\to 0 $
. There is no restriction on the decay rates of
$ V $
. Especially,
$ V $
can be compactly supported. The appearance of
$ A $
and the nonlocal of
$ (-\Delta)^s $
makes the proof more difficult than that in [7], which considered the case
$ A\equiv 0 $
.
Citation: Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038
References:
[1]

C. O. Alves and O. H. Miyagaki, 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), Art.47, 19pp. doi: 10.1007/s00526-016-0983-x.

[2]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., 235 (2003), 427-466.  doi: 10.1007/s00220-003-0811-y.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[4]

V. Ambrosio, A local mountain pass approach for a class of fractional NLS equations with magnetic fields, Nonlinear Anal., 190 (2020), 111622, 14pp. doi: 10.1016/j.na.2019.111622.

[5]

V. Ambrosio, Existence and concentration results for some fractional Schrödinger equations in ${\mathbb R}^N$ with magnitic fields, Commun. Partial Differ. Equ., 44 (2019), 637-680.  doi: 10.1080/03605302.2019.1581800.

[6]

V. Ambrosio and P. d'Avenia, Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differ. Equ., 264 (2018), 3336-3368.  doi: 10.1016/j.jde.2017.11.021.

[7]

X. AnL. Duan and Y. Peng, Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials., Appl. Anal., (2021), 1-18.  doi: 10.1080/00036811.2021.1880571.

[8]

X. An, S. Peng and C. Xie, Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity, J. Math. Phys., 60 (2019), 021501, 18pp. doi: 10.1063/1.5037126.

[9]

D. Bonheure, S. Cingolani and M. Nys, Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic filed, Calc. Var. Partial Differ. Equ., 55 (2016), 33pp. doi: 10.1007/s00526-016-1013-8.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[11]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[12]

P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var., 24 (2018), 1-24.  doi: 10.1051/cocv/2016071.

[13]

J. DávilaM. del PinoS. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum, Anal. Partial Differ. Equ., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.

[14]

J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[15]

M. del Pino and P. L. Felmer, Local Mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.

[16]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.

[17]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.

[18]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[19]

M. M. FallF. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[20]

P. FelmerA. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[21]

A. FiscellaA. Pinamonti and E. Vecchi, Multiplicity results for magnetic fractional problems, J. Differ. Equ., 263 (2017), 4617-4633.  doi: 10.1016/j.jde.2017.05.028.

[22]

L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacians, Commun. Pure. Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[23]

L. Frank and R. Seiringer, Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[24]

T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, 2013. doi: 10.1007/978-3-0348-0591-9_5.

[25]

K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal.: Theory, Methods and Applications, 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3.

[26]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.

[27]

A. PinamontiM. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., 12 (2017), 225-252.  doi: 10.1515/acv-2017-0019.

[28]

A. PinamontiM. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.  doi: 10.1016/j.jmaa.2016.12.065.

[29]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb R}^N$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[30]

M. Squassina and B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, 354 (2016), 825-831.  doi: 10.1016/j.crma.2016.04.013.

[31]

M. SquassinaB. Zhang and X. Zhang, Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscr. Math., 155 (2018), 115-140.  doi: 10.1007/s00229-017-0937-4.

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

C. O. Alves and O. H. Miyagaki, 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), Art.47, 19pp. doi: 10.1007/s00526-016-0983-x.

[2]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., 235 (2003), 427-466.  doi: 10.1007/s00220-003-0811-y.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[4]

V. Ambrosio, A local mountain pass approach for a class of fractional NLS equations with magnetic fields, Nonlinear Anal., 190 (2020), 111622, 14pp. doi: 10.1016/j.na.2019.111622.

[5]

V. Ambrosio, Existence and concentration results for some fractional Schrödinger equations in ${\mathbb R}^N$ with magnitic fields, Commun. Partial Differ. Equ., 44 (2019), 637-680.  doi: 10.1080/03605302.2019.1581800.

[6]

V. Ambrosio and P. d'Avenia, Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differ. Equ., 264 (2018), 3336-3368.  doi: 10.1016/j.jde.2017.11.021.

[7]

X. AnL. Duan and Y. Peng, Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials., Appl. Anal., (2021), 1-18.  doi: 10.1080/00036811.2021.1880571.

[8]

X. An, S. Peng and C. Xie, Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity, J. Math. Phys., 60 (2019), 021501, 18pp. doi: 10.1063/1.5037126.

[9]

D. Bonheure, S. Cingolani and M. Nys, Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic filed, Calc. Var. Partial Differ. Equ., 55 (2016), 33pp. doi: 10.1007/s00526-016-1013-8.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[11]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[12]

P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var., 24 (2018), 1-24.  doi: 10.1051/cocv/2016071.

[13]

J. DávilaM. del PinoS. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum, Anal. Partial Differ. Equ., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.

[14]

J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[15]

M. del Pino and P. L. Felmer, Local Mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.

[16]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.

[17]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.

[18]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[19]

M. M. FallF. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[20]

P. FelmerA. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[21]

A. FiscellaA. Pinamonti and E. Vecchi, Multiplicity results for magnetic fractional problems, J. Differ. Equ., 263 (2017), 4617-4633.  doi: 10.1016/j.jde.2017.05.028.

[22]

L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacians, Commun. Pure. Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[23]

L. Frank and R. Seiringer, Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[24]

T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, 2013. doi: 10.1007/978-3-0348-0591-9_5.

[25]

K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal.: Theory, Methods and Applications, 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3.

[26]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.

[27]

A. PinamontiM. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., 12 (2017), 225-252.  doi: 10.1515/acv-2017-0019.

[28]

A. PinamontiM. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.  doi: 10.1016/j.jmaa.2016.12.065.

[29]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb R}^N$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[30]

M. Squassina and B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, 354 (2016), 825-831.  doi: 10.1016/j.crma.2016.04.013.

[31]

M. SquassinaB. Zhang and X. Zhang, Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscr. Math., 155 (2018), 115-140.  doi: 10.1007/s00229-017-0937-4.

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

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