American Institute of Mathematical Sciences

Advanced
November  2018, 38(11): 5389-5413. doi: 10.3934/dcds.2018238

On the concentration of semiclassical states for nonlinear Dirac equations

Xu Zhang ,

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author: Xu Zhang

Received  July 2017 Revised  December 2017 Published  August 2018

Fund Project: The first author is supported by the China Postdoctoral Science Foundation 2017M611160.

In this paper, we study the following nonlinear Dirac equation
$\begin{equation*}-i\varepsilonα·\nabla w+aβ w+V(x)w = g(|w|)w, \ x∈ \mathbb{R}^3, \ {\rm for}\ w∈ H^1(\mathbb R^3, \mathbb C^4), \end{equation*}$
where
$a > 0$
 is a constant,
$α = (α_1, α_2, α_3)$
,
$α_1, α_2, α_3$
 and
$β$
 are
$4×4$
 Pauli-Dirac matrices. Under the assumptions that
$V$
 and
$g$
 are continuous but are not necessarily of class
$C^1$
, when
$g$
 is super-linear growth at infinity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as
$\varepsilon \to 0$
.
Keywords: Dirac equation, semiclassical solutions, concentration.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.
Citation: Xu Zhang. On the concentration of semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5389-5413. doi: 10.3934/dcds.2018238
References:
[1]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres Ⅰ, Comm. Math. Phys., 235 (2003), 427-466.  doi: 10.1007/s00220-003-0811-y.  Google Scholar

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T. Bartsch and Y. H. Ding, Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.  doi: 10.1016/j.jde.2005.08.014.  Google Scholar

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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

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J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. Ⅱ, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.  doi: 10.1007/s00526-002-0191-8.  Google Scholar

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P. D'Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the onlinear 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

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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.  Google Scholar

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M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

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M. Del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

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

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Y. H. 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.  Google Scholar

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Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 765-790.  doi: 10.1017/S0308210511001752.  Google Scholar

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Y. H. 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.  Google Scholar

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Y. H. 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.  Google Scholar

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Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, J. Math. Phys., 20 (2008), 1007-1032.  doi: 10.1142/S0129055X0800350X.  Google Scholar

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

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M.J. Esteban, M. Lewin and E. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 535-593.  doi: 10.1090/S0273-0979-08-01212-3.  Google Scholar

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G. M. Figueiredo and M. T. O. Pimenta, Existence of ground state solutions to Dirac equations with vanishing potentials at infinity, J. Differential Equations, 262 (2017), 486-505.  doi: 10.1016/j.jde.2016.09.034.  Google Scholar

[22]

A. Floer and A. Weistein, Nonspreading wave pockets 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

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1997.  Google Scholar

[24]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318.  doi: 10.1007/s00526-003-0261-6.  Google Scholar

[25]

X. Kang and J. C. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928.   Google Scholar

[26]

Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Diff. Equ., 2 (1997), 955-980.   Google Scholar

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[28]

S. B. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  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.  Google Scholar

[30]

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

[31]

A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7.  Google Scholar

[32]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-292.  doi: 10.1007/BF00946631.  Google Scholar

[33]

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.  Google Scholar

[34]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[35]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.  doi: 10.1137/S0036141095290240.  Google Scholar

[36]

J. Zhang, X. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys., 54 (2013), 101502, 10pp. doi: 10.1063/1.4824132.  Google Scholar

[37]

J. Zhang, X. Tang and W. Zhang, On ground state solutions for superlinear Dirac equation, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 840-850.  doi: 10.1016/S0252-9602(14)60054-0.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres Ⅰ, Comm. Math. Phys., 235 (2003), 427-466.  doi: 10.1007/s00220-003-0811-y.  Google Scholar

[2]

T. Bartsch and Y. H. Ding, Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.  doi: 10.1016/j.jde.2005.08.014.  Google Scholar

[3]

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

[4]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. Ⅱ, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.  doi: 10.1007/s00526-002-0191-8.  Google Scholar

[5]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3, Springer, Berlin, 1990.  Google Scholar

[6]

P. D'Avenia, A. Pomponio and D. Ruiz, Semiclassical states for the onlinear 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

[7]

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.  Google Scholar

[8]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[9]

M. Del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

[10]

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

[11]

Y. H. 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.  Google Scholar

[12]

Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 765-790.  doi: 10.1017/S0308210511001752.  Google Scholar

[13]

Y. H. Ding and X. Y. 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.  Google Scholar

[14]

Y. H. 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.  Google Scholar

[15]

Y. H. 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.  Google Scholar

[16]

Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, J. Math. Phys., 20 (2008), 1007-1032.  doi: 10.1142/S0129055X0800350X.  Google Scholar

[17]

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

[18]

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

[19]

M.J. Esteban, M. Lewin and E. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 535-593.  doi: 10.1090/S0273-0979-08-01212-3.  Google Scholar

[20]

M. J. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Commun. Math. Phys., 171 (1995), 323-350.  doi: 10.1007/BF02099273.  Google Scholar

[21]

G. M. Figueiredo and M. T. O. Pimenta, Existence of ground state solutions to Dirac equations with vanishing potentials at infinity, J. Differential Equations, 262 (2017), 486-505.  doi: 10.1016/j.jde.2016.09.034.  Google Scholar

[22]

A. Floer and A. Weistein, Nonspreading wave pockets 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

[23]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1997.  Google Scholar

[24]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318.  doi: 10.1007/s00526-003-0261-6.  Google Scholar

[25]

X. Kang and J. C. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928.   Google Scholar

[26]

Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Diff. Equ., 2 (1997), 955-980.   Google Scholar

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[28]

S. B. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  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.  Google Scholar

[30]

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

[31]

A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7.  Google Scholar

[32]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-292.  doi: 10.1007/BF00946631.  Google Scholar

[33]

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.  Google Scholar

[34]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[35]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.  doi: 10.1137/S0036141095290240.  Google Scholar

[36]

J. Zhang, X. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys., 54 (2013), 101502, 10pp. doi: 10.1063/1.4824132.  Google Scholar

[37]

J. Zhang, X. Tang and W. Zhang, On ground state solutions for superlinear Dirac equation, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 840-850.  doi: 10.1016/S0252-9602(14)60054-0.  Google Scholar

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