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

Previous Article
Analysis of one-sided 1-D fractional diffusion operator
CPAA Home
This Issue
Next Article
Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

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

Xiaoming An ^1, and Xian Yang ^2,,

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 $

.

Keywords: Fractional magnetic Schrödinger, compactly supported, penalized, variational, concentration.

Mathematics Subject Classification: 35J05, 35A15, 35J10.

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. Ambrosetti, A. 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. An, L. 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. Dávila, M. del Pino, S. 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. Dávila, M. 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 Pino, M. 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 Nezza, G. 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. Fall, F. 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. Felmer, A. 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. Fiscella, A. 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. Frank, E. 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. Pinamonti, M. 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. Pinamonti, M. 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. Squassina, B. 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. Ambrosetti, A. 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. An, L. 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. Dávila, M. del Pino, S. 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. Dávila, M. 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 Pino, M. 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 Nezza, G. 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. Fall, F. 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. Felmer, A. 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. Fiscella, A. 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. Frank, E. 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. Pinamonti, M. 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. Pinamonti, M. 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. Squassina, B. 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.

[1]	Yosra Soussi. Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide. Inverse Problems and Imaging, 2021, 15 (5) : 929-950. doi: 10.3934/ipi.2021022
[2]	Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
[3]	Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
[4]	Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[5]	Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254
[6]	César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure and Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535
[7]	Vincenzo Ambrosio. The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5659-5705. doi: 10.3934/dcds.2021092
[8]	Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014
[9]	Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80
[10]	Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
[11]	Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[12]	Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure and Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281
[13]	Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457
[14]	Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209
[15]	Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039
[16]	Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067
[17]	Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[18]	Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[19]	Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[20]	Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems and Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Xiaoming An Xian Yang
on Google Scholar
Xiaoming An Xian Yang

2021 Impact Factor: 1.273

Tools

Article outline

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Xiaoming An Xian Yang
on Google Scholar
Xiaoming An Xian Yang

[Back to Top]