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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 |
$ 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*} $ |
$ n\geq 2 $ |
$ \hbar>0 $ |
$ a>0 $ |
$ f $ |
$ W(x)|u|^{p-2} $ |
$ W_{1}(x)|u|^{p-2}+W_{2}(x)|u|^{2^*-2} $ |
$ p\in (2,2^*), 2^* = \frac{2n}{n-1} $ |
$ \min V $ |
$ \liminf\limits_{|x|\rightarrow \infty} V(x) $ |
$ \max W $ |
$ \limsup\limits_{|x|\rightarrow \infty} W(x) $ |
$ \max W_{j} $ |
$ \limsup\limits_{|x|\rightarrow \infty} W_{j}(x), j = 1,2, $ |
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). 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. |
[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. 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. |
[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. |
[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). 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. |
[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. 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. |
[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. |
[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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