doi: 10.3934/dcds.2021030

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

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  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.
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 - A, 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. AmbrosettiM. 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. BalabaneT. CazenaveA. 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. BalabaneT. 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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Y. ChenY. 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'AnconaD. 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, 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

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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

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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

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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

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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

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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. EstebanM. 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

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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. FinkelsteinC. 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. AmbrosettiM. 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. BalabaneT. CazenaveA. 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. BalabaneT. 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. ChenY. 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'AnconaD. 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. EstebanM. 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. FinkelsteinC. 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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