American Institute of Mathematical Sciences

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January  2021, 41(1): 29-60. doi: 10.3934/dcds.2020297

Strongly localized semiclassical states for nonlinear Dirac equations

Thomas Bartsch 1,, and  Tian Xu 2,

1. 

Mathematisches Institut, Universität Giessen, 35392, Giessen, Germany
2. 

Center for Applied Mathematics, Tianjin University, 300072, Tianjin, China

* Corresponding author: Thomas Bartsch

Received  January 2020 Revised  May 2020 Published  January 2021 Early access  August 2020

Fund Project: The second author is supported by the National Science Foundation of China (NSFC 11601370, 11771325) and the Alexander von Humboldt Foundation of Germany

We study semiclassical states of the nonlinear Dirac equation
$ -i\hbar{\partial}_t\psi = ic\hbar\sum\limits_{k = 1}^3{\alpha}_k{\partial}_k\psi - mc^2{\beta} \psi - M(x)\psi + f(|\psi|)\psi, \quad t\in \mathbb{R}, \ x\in \mathbb{R}^3, $
where
$ V $
 is a bounded continuous potential function and the nonlinear term
$ f(|\psi|)\psi $
 is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity
$ f(s) = s^p $
. We develop a variational method for the strongly indefinite functional associated to the problem.
Keywords: Dirac equation, semiclassical states, standing waves, concentration, strongly indefinite functional.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.
Citation: Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
References:
[1]

N. Ackermann, A nonlinear superposition principle and multibump solution 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. Cignolani, Semi-classical states of nonlinear Shrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[4]

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Math. Ann., 338 (2007), 147-185.  doi: 10.1007/s00208-006-0071-1.  Google Scholar

[5]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[6]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.  doi: 10.1007/s00205-002-0225-6.  Google Scholar

[7]

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

[8]

P. d'Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.  doi: 10.1016/j.jfa.2012.03.009.  Google Scholar

[9]

M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[10]

M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis., 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[11]

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

[12]

Y. Ding, Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., 7, World Scientific Publ., 2007. doi: 10.1142/9789812709639.  Google Scholar

[13]

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

[14]

Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 765-790.  doi: 10.1017/S0308210511001752.  Google Scholar

[15]

Y. Ding, Z. Liu and J. Wei, Multiplicity and concentration of semi-classical solutions to nonlinear Dirac equations, 2017. Available from: http://www.math.ubc.ca/ jcwei/MulDirac-2017-03-13.pdf. Google Scholar

[16]

Y. Ding and B. Ruf, Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities, SIAM Journal on Mathematical Analysis, 44 (2012), 3755-3785.  doi: 10.1137/110850670.  Google Scholar

[17]

Y. Ding, J. Wei and T. Xu, Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys., 54 (2013), 061505, 33 pp. doi: 10.1063/1.4811541.  Google Scholar

[18]

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

[19]

Y. Ding and T. Xu, Concentrating patterns of reaction-diffusion systems: A variational approach, Trans. Amer. Math. Soc., 369 (2017), 97-138.  doi: 10.1090/tran/6626.  Google Scholar

[20]

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

[21]

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

[22]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326-332.  doi: 10.1103/PhysRev.83.326.  Google Scholar

[23]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571-1579.  doi: 10.1103/PhysRev.103.1571.  Google Scholar

[24]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[25]

W. L. Fushchich and R. Z. Zhdanov, Symmetry and exact solutions of nonlinear spinor equations, Phys. Rep., 172 (1989), 123-174.  doi: 10.1016/0370-1573(89)90090-2.  Google Scholar

[26]

W. Fushchich and R. Zhdanov, Symmetries and Exact Solutions of Nonlinear Dirac Equations, Mathematical Ukraina Publisher, Kyiv, 1997. Google Scholar

[27]

L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[28]

L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries, Phys. D, 238 (2009), 1413-1421.  doi: 10.1016/j.physd.2009.02.001.  Google Scholar

[29]

L. H. Haddad, C. M. Weaver and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices, New J. Phys., 17 (2015), 063033, 23pp. doi: 10.1088/1367-2630/17/6/063033.  Google Scholar

[30]

L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: II. Relativistic soliton stability analysis, New J. Phys., 17 (2015), 063034, 22pp. doi: 10.1088/1367-2630/17/6/063034.  Google Scholar

[31]

D. D. Ivanenko, Notes to the theory of interaction via particles, Zh.Éksp. Teor. Fiz., 8 (1938), 260-266.   Google Scholar

[32]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ \mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[33]

J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78. doi: 10.24033/asens.836.  Google Scholar

[34]

P.-L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, Part II, Annales de l'Institut Henri Poincare (C) Non Linear Analysis., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[35]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200.  doi: 10.12775/TMNA.1997.008.  Google Scholar

[36]

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

[37]

W. K. Ng and R. R. Parwani, Nonlinear Dirac Equations, Symm. Integr. Geom. Method. Appl. 5 (2009), 023, 20 pages. doi: 10.3842/SIGMA.2009.023.  Google Scholar

[38]

Y.-G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[39]

Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[40]

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

[41] M. Reed and B. Simon, Methods of Mathematical Physics, Vols. I-IV, Academic Press, 1978.   Google Scholar
[42]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632.  Google Scholar

[43]

F. M. Toyama, Y. Hosono, B. Ilyas and Y. Nogami, Reduction of the nonlinear Dirac equation to a nonlinear Schrödinger equation with a correction term, J. Phys. A, 27 (1994), 3139-3148.  doi: 10.1088/0305-4470/27/9/026.  Google Scholar

[44]

Z.-Q. 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 pp. doi: 10.1007/s00526-018-1319-9.  Google Scholar

[45]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. 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 solution 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. Cignolani, Semi-classical states of nonlinear Shrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.  doi: 10.1007/s002050050067.  Google Scholar

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[4]

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Math. Ann., 338 (2007), 147-185.  doi: 10.1007/s00208-006-0071-1.  Google Scholar

[5]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[6]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.  doi: 10.1007/s00205-002-0225-6.  Google Scholar

[7]

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

[8]

P. d'Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.  doi: 10.1016/j.jfa.2012.03.009.  Google Scholar

[9]

M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[10]

M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis., 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[11]

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

[12]

Y. Ding, Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., 7, World Scientific Publ., 2007. doi: 10.1142/9789812709639.  Google Scholar

[13]

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

[14]

Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 765-790.  doi: 10.1017/S0308210511001752.  Google Scholar

[15]

Y. Ding, Z. Liu and J. Wei, Multiplicity and concentration of semi-classical solutions to nonlinear Dirac equations, 2017. Available from: http://www.math.ubc.ca/ jcwei/MulDirac-2017-03-13.pdf. Google Scholar

[16]

Y. Ding and B. Ruf, Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities, SIAM Journal on Mathematical Analysis, 44 (2012), 3755-3785.  doi: 10.1137/110850670.  Google Scholar

[17]

Y. Ding, J. Wei and T. Xu, Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys., 54 (2013), 061505, 33 pp. doi: 10.1063/1.4811541.  Google Scholar

[18]

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

[19]

Y. Ding and T. Xu, Concentrating patterns of reaction-diffusion systems: A variational approach, Trans. Amer. Math. Soc., 369 (2017), 97-138.  doi: 10.1090/tran/6626.  Google Scholar

[20]

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

[21]

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

[22]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326-332.  doi: 10.1103/PhysRev.83.326.  Google Scholar

[23]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571-1579.  doi: 10.1103/PhysRev.103.1571.  Google Scholar

[24]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[25]

W. L. Fushchich and R. Z. Zhdanov, Symmetry and exact solutions of nonlinear spinor equations, Phys. Rep., 172 (1989), 123-174.  doi: 10.1016/0370-1573(89)90090-2.  Google Scholar

[26]

W. Fushchich and R. Zhdanov, Symmetries and Exact Solutions of Nonlinear Dirac Equations, Mathematical Ukraina Publisher, Kyiv, 1997. Google Scholar

[27]

L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[28]

L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries, Phys. D, 238 (2009), 1413-1421.  doi: 10.1016/j.physd.2009.02.001.  Google Scholar

[29]

L. H. Haddad, C. M. Weaver and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices, New J. Phys., 17 (2015), 063033, 23pp. doi: 10.1088/1367-2630/17/6/063033.  Google Scholar

[30]

L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Bose-Einstein condensates: II. Relativistic soliton stability analysis, New J. Phys., 17 (2015), 063034, 22pp. doi: 10.1088/1367-2630/17/6/063034.  Google Scholar

[31]

D. D. Ivanenko, Notes to the theory of interaction via particles, Zh.Éksp. Teor. Fiz., 8 (1938), 260-266.   Google Scholar

[32]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ \mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[33]

J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78. doi: 10.24033/asens.836.  Google Scholar

[34]

P.-L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, Part II, Annales de l'Institut Henri Poincare (C) Non Linear Analysis., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[35]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200.  doi: 10.12775/TMNA.1997.008.  Google Scholar

[36]

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

[37]

W. K. Ng and R. R. Parwani, Nonlinear Dirac Equations, Symm. Integr. Geom. Method. Appl. 5 (2009), 023, 20 pages. doi: 10.3842/SIGMA.2009.023.  Google Scholar

[38]

Y.-G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar

[39]

Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar

[40]

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

[41] M. Reed and B. Simon, Methods of Mathematical Physics, Vols. I-IV, Academic Press, 1978.   Google Scholar
[42]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632.  Google Scholar

[43]

F. M. Toyama, Y. Hosono, B. Ilyas and Y. Nogami, Reduction of the nonlinear Dirac equation to a nonlinear Schrödinger equation with a correction term, J. Phys. A, 27 (1994), 3139-3148.  doi: 10.1088/0305-4470/27/9/026.  Google Scholar

[44]

Z.-Q. 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 pp. doi: 10.1007/s00526-018-1319-9.  Google Scholar

[45]

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

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