Advanced Search
Article Contents
Article Contents

Strongly localized semiclassical states for nonlinear Dirac equations

  • * Corresponding author: Thomas Bartsch

    * Corresponding author: Thomas Bartsch 
The second author is supported by the National Science Foundation of China (NSFC 11601370, 11771325) and the Alexander von Humboldt Foundation of Germany
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. AmbrosettiM. 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. AmbrosettiV. 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. BartschM. 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'AveniaA. 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. DingC. 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. FinkelsteinR. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326-332.  doi: 10.1103/PhysRev.83.326.
    [23] R. FinkelsteinC. 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. SimonMethods 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. ToyamaY. HosonoB. 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.
  • 加载中

Article Metrics

HTML views(1395) PDF downloads(261) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint