# American Institute of Mathematical Sciences

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

## Strongly localized semiclassical states for nonlinear Dirac equations

 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.
Citation: Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. Ding, Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., 7, World Scientific Publ., 2007. doi: 10.1142/9789812709639. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326-332.  doi: 10.1103/PhysRev.83.326. [23] R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571-1579.  doi: 10.1103/PhysRev.103.1571. [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. [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. [26] W. Fushchich and R. Zhdanov, Symmetries and Exact Solutions of Nonlinear Dirac Equations, Mathematical Ukraina Publisher, Kyiv, 1997. [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. [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. [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. [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. [31] D. D. Ivanenko, Notes to the theory of interaction via particles, Zh.Éksp. Teor. Fiz., 8 (1938), 260-266. [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. [33] J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78. doi: 10.24033/asens.836. [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. [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. [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. [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. [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. [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. [40] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys, 43 (1992), 270-291.  doi: 10.1007/BF00946631. [41] M. Reed and B. Simon, Methods of Mathematical Physics, Vols. I-IV, Academic Press, 1978. [42] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. Ding, Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., 7, World Scientific Publ., 2007. doi: 10.1142/9789812709639. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326-332.  doi: 10.1103/PhysRev.83.326. [23] R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571-1579.  doi: 10.1103/PhysRev.103.1571. [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. [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. [26] W. Fushchich and R. Zhdanov, Symmetries and Exact Solutions of Nonlinear Dirac Equations, Mathematical Ukraina Publisher, Kyiv, 1997. [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. [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. [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. [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. [31] D. D. Ivanenko, Notes to the theory of interaction via particles, Zh.Éksp. Teor. Fiz., 8 (1938), 260-266. [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. [33] J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78. doi: 10.24033/asens.836. [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. [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. [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. [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. [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. [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. [40] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys, 43 (1992), 270-291.  doi: 10.1007/BF00946631. [41] M. Reed and B. Simon, Methods of Mathematical Physics, Vols. I-IV, Academic Press, 1978. [42] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632. [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. [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. [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.
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