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September  2021, 41(9): 4105-4123. doi: 10.3934/dcds.2021030

On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $

Yanheng Ding , , Xiaojing Dong and  Qi Guo

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China, University of Chinese Academy of Sciences, Beijing 100049, P. R. China

* Corresponding author: dingyh@math.ac.cn

Received  October 2020 Revised  December 2020 Published  September 2021 Early access  January 2021

Fund Project: The authors are supported by NSFC 11871242, 11801545

In this paper, we study multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension
$ n $
:
$ \begin{equation*} -i\hbar\sum\limits_{k = 1}^n \alpha_k \partial_k u+a\beta u+V(x)u = f(x,|u|)u,\; \text{in}\ \mathbb{R}^n, \end{equation*} $
where
$ n\geq 2 $
,
$ \hbar>0 $
 is a small parameter,
$ a>0 $
 is a constant, and
$ f $
 describes the self-interaction which is either subcritical:
$ W(x)|u|^{p-2} $
, or critical:
$ W_{1}(x)|u|^{p-2}+W_{2}(x)|u|^{2^*-2} $
, with
$ p\in (2,2^*), 2^* = \frac{2n}{n-1} $
. The number of solutions obtained depending on the ratio of
$ \min V $
 and
$ \liminf\limits_{|x|\rightarrow \infty} V(x) $
, as well as
$ \max W $
 and
$ \limsup\limits_{|x|\rightarrow \infty} W(x) $
 for the subcritical case and
$ \max W_{j} $
 and
$ \limsup\limits_{|x|\rightarrow \infty} W_{j}(x), j = 1,2, $
 for the critical case.
Keywords: Nonlinear Dirac equation, Semi-classical solutions, Multiplicity, Variational methods, subcritical and critical nonlinearity.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.
Citation: Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030
References:
[1]

N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.  doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

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M. Balabane, T. Cazenave, A. Douady and F. Merle, Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys., 119 (1988), 153-176.  doi: 10.1007/BF01218265.  Google Scholar

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M. Balabane, T. Cazenave and L. Vázquez, Existence of standing waves for Dirac fields with singular nonlinearities, Comm. Math. Phys., 133 (1990), 53-74.  doi: 10.1007/BF02096554.  Google Scholar

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T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279 (2006), 1267-1288.   Google Scholar

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T. Bartsch and Y. Ding, Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.  doi: 10.1016/j.jde.2005.08.014.  Google Scholar

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N. Bournaveas, Low regularity solutions of the dirac klein-gordon equations in two space dimensions, Communications in Partial Differential Equations, 26 (2001), 1245-1266.  doi: 10.1081/PDE-100106136.  Google Scholar

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T. Cazenave and L. Vázquez, Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 105 (1986), 35-47.  doi: 10.1007/BF01212340.  Google Scholar

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Y. Chen, Y. Ding and T. Xu, Potential well and multiplicity of solutions for nonlinear dirac equations, Commun. Pure Appl. Anal., 19 (2020), 587-607.  doi: 10.3934/cpaa.2020028.  Google Scholar

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S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

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P. D'Ancona, D. Foschi and S. Selberg, Local well-posedness below the charge norm for the dirac-klein-gordon system in two space dimensions, Journal of Hyperbolic Differential Equations, 4 (2007), 295-330.  doi: 10.1142/S0219891607001148.  Google Scholar

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

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M. Del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

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Y. Ding, Variational Methods for Strongly Indefinite Problems, volume 7 of Interdisciplinary Mathematical Sciences, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.  Google Scholar

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Y. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.  doi: 10.1016/j.jde.2010.03.022.  Google Scholar

[16]

Y. Ding, Q. Guo and T. Xu, Concentration of semi-classical states for nonlinear dirac equations of space-dimension $n$, Minimax Theory Appl., 6 (2021). Google Scholar

[17]

Y. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.  doi: 10.1016/j.jde.2012.01.023.  Google Scholar

[18]

Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.  doi: 10.1007/s00229-011-0530-1.  Google Scholar

[19]

Y. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82.  doi: 10.1007/s00205-008-0163-z.  Google Scholar

[20]

Y. Ding and B. Ruf, Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.  doi: 10.1137/110850670.  Google Scholar

[21]

Y. Ding and J. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032.  doi: 10.1142/S0129055X0800350X.  Google Scholar

[22]

Y. Ding and T. Xu, Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Ration. Mech. Anal., 216 (2015), 415-447.  doi: 10.1007/s00205-014-0811-4.  Google Scholar

[23]

M. J. Esteban, M. Lewin and E. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., 45 (2008), 535-593.  doi: 10.1090/S0273-0979-08-01212-3.  Google Scholar

[24]

M. J. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys., 171 (1995), 323-350.  doi: 10.1007/BF02099273.  Google Scholar

[25]

M. J. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst., Current developments in partial differential equations (Temuco, 1999), 8 (2002), 381–397. doi: 10.3934/dcds.2002.8.381.  Google Scholar

[26]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review., 103 (1956), 1571-1579.   Google Scholar

[27]

A. Grünrock and H. Pecher, Global solutions for the dirac klein gordon system in two space dimensions, Communications in Partial Differential Equations, 35 (2010), 89-112.  doi: 10.1080/03605300903296306.  Google Scholar

[28]

D. Ivanenko, Notes to the theory of interaction via particles, Zhurn. Experim.Teoret. Fiz., 8 (1938), 260-266.   Google Scholar

[29]

F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74 (1988), 50-68.  doi: 10.1016/0022-0396(88)90018-6.  Google Scholar

[30]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[31]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[32]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[33]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

[34]

Z. Wang and X. Zhang, An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 56, 30. doi: 10.1007/s00526-018-1319-9.  Google Scholar

[35]

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

show all references

References:
[1]

N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.  doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[3]

M. Balabane, T. Cazenave, A. Douady and F. Merle, Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys., 119 (1988), 153-176.  doi: 10.1007/BF01218265.  Google Scholar

[4]

M. Balabane, T. Cazenave and L. Vázquez, Existence of standing waves for Dirac fields with singular nonlinearities, Comm. Math. Phys., 133 (1990), 53-74.  doi: 10.1007/BF02096554.  Google Scholar

[5]

T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279 (2006), 1267-1288.   Google Scholar

[6]

T. Bartsch and Y. Ding, Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.  doi: 10.1016/j.jde.2005.08.014.  Google Scholar

[7]

N. Bournaveas, Low regularity solutions of the dirac klein-gordon equations in two space dimensions, Communications in Partial Differential Equations, 26 (2001), 1245-1266.  doi: 10.1081/PDE-100106136.  Google Scholar

[8]

T. Cazenave and L. Vázquez, Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 105 (1986), 35-47.  doi: 10.1007/BF01212340.  Google Scholar

[9]

Y. Chen, Y. Ding and T. Xu, Potential well and multiplicity of solutions for nonlinear dirac equations, Commun. Pure Appl. Anal., 19 (2020), 587-607.  doi: 10.3934/cpaa.2020028.  Google Scholar

[10]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

[11]

P. D'Ancona, D. Foschi and S. Selberg, Local well-posedness below the charge norm for the dirac-klein-gordon system in two space dimensions, Journal of Hyperbolic Differential Equations, 4 (2007), 295-330.  doi: 10.1142/S0219891607001148.  Google Scholar

[12]

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

[13]

M. Del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

[14]

Y. Ding, Variational Methods for Strongly Indefinite Problems, volume 7 of Interdisciplinary Mathematical Sciences, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.  Google Scholar

[15]

Y. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.  doi: 10.1016/j.jde.2010.03.022.  Google Scholar

[16]

Y. Ding, Q. Guo and T. Xu, Concentration of semi-classical states for nonlinear dirac equations of space-dimension $n$, Minimax Theory Appl., 6 (2021). Google Scholar

[17]

Y. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.  doi: 10.1016/j.jde.2012.01.023.  Google Scholar

[18]

Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.  doi: 10.1007/s00229-011-0530-1.  Google Scholar

[19]

Y. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82.  doi: 10.1007/s00205-008-0163-z.  Google Scholar

[20]

Y. Ding and B. Ruf, Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.  doi: 10.1137/110850670.  Google Scholar

[21]

Y. Ding and J. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032.  doi: 10.1142/S0129055X0800350X.  Google Scholar

[22]

Y. Ding and T. Xu, Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Ration. Mech. Anal., 216 (2015), 415-447.  doi: 10.1007/s00205-014-0811-4.  Google Scholar

[23]

M. J. Esteban, M. Lewin and E. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., 45 (2008), 535-593.  doi: 10.1090/S0273-0979-08-01212-3.  Google Scholar

[24]

M. J. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys., 171 (1995), 323-350.  doi: 10.1007/BF02099273.  Google Scholar

[25]

M. J. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst., Current developments in partial differential equations (Temuco, 1999), 8 (2002), 381–397. doi: 10.3934/dcds.2002.8.381.  Google Scholar

[26]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review., 103 (1956), 1571-1579.   Google Scholar

[27]

A. Grünrock and H. Pecher, Global solutions for the dirac klein gordon system in two space dimensions, Communications in Partial Differential Equations, 35 (2010), 89-112.  doi: 10.1080/03605300903296306.  Google Scholar

[28]

D. Ivanenko, Notes to the theory of interaction via particles, Zhurn. Experim.Teoret. Fiz., 8 (1938), 260-266.   Google Scholar

[29]

F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74 (1988), 50-68.  doi: 10.1016/0022-0396(88)90018-6.  Google Scholar

[30]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[31]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[32]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[33]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

[34]

Z. Wang and X. Zhang, An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 56, 30. doi: 10.1007/s00526-018-1319-9.  Google Scholar

[35]

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

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