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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 and 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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[33]

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

[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.

[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.

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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[33]

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

[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.

[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.

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