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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 |
$ -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, $ |
$ V $ |
$ f(|\psi|)\psi $ |
$ f(s) = s^p $ |
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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