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January  2020, 19(1): 587-607. doi: 10.3934/cpaa.2020028

Potential well and multiplicity of solutions for nonlinear Dirac equations

Yu Chen 1, , Yanheng Ding 2, and  Tian Xu 3,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex System, Ministry of Education, 100875 Beijing, China
2. 

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China
3. 

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

Received  September 2018 Revised  September 2018 Published  July 2019

In this paper we consider the semi-classical solutions of a massive Dirac equations in presence of a critical growth nonlinearity
$ -i\hbar \sum\limits_{k = 1}^{3}\alpha_k\partial_k w+a\beta w+V(x)w = f(|w|)w. $
Under a local condition imposed on the potential
$ V $
, we relate the number of solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.
Keywords: Dirac equations, semi-classical states, multiplicity.
Mathematics Subject Classification: 35Q40, 49J35.
Citation: Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028
References:
[1]

N. Ackermann, A nonlinear superposition principle and multibump solution of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 423-443.  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]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159, (2001), 253–271. doi: 10.1007/s002050100152.

[5]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[6]

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.

[7]

V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.  doi: 10.1007/BF00375686.

[8]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314.

[9]

V. Benci, G. Cerami and D. Passaseo, On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear Analysis, tribute in honor of G. Prodi, Quaderno Scuola Normale Sup. Pisa, (1991), 93–107.

[10]

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.

[11]

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.

[12]

G. Cerami and D. Passaseo, Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with "rich" topology, Nonlinear Anal. TMA, 18 (1992), 109-119.  doi: 10.1016/0362-546X(92)90089-W.

[13]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[14]

S. Cingolani, L. Jeanjean and K. Tanaka, Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439.  doi: 10.1007/s00526-014-0754-5.

[15]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differ. Equ., 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.

[16]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem, Top. Methods Nonlinear Anal., 4 (1999), 347-368.  doi: 10.12775/TMNA.1998.016.

[17]

E. N. Dancer and S. Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Differ. Equ., 4 (1999), 347–368.

[18]

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

[19]

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.

[20]

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

[21]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 15. No. 2. Elsevier Masson, 1998, 127–149. doi: 10.1016/S0294-1449(97)89296-7.

[22]

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.

[23]

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

[24]

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

[25]

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.

[26]

Y. H. 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.

[27]

Y. H. 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.

[28]

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

[29]

J. Esteban and Eric Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 323–350.

[30]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326–332.

[31]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571–1579.

[32]

G. Fournier and M. Willem, Relative category and the calculus of variations, in Variational Methods, H. Berestycki et al. Birkhäuser Boston, 4 (1990), 95–104.

[33]

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.

[34]

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

[35]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, Part Ⅱ, AIP Anal. non linéaire, 1, 223–283.

[36]

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.

[37]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223–253.

[38]

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

[39]

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.

[40]

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.

[41]

M. Willem, Minimax Theorems, Birkhäuser, 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), 423-443.  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]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159, (2001), 253–271. doi: 10.1007/s002050100152.

[5]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[6]

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.

[7]

V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.  doi: 10.1007/BF00375686.

[8]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314.

[9]

V. Benci, G. Cerami and D. Passaseo, On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear Analysis, tribute in honor of G. Prodi, Quaderno Scuola Normale Sup. Pisa, (1991), 93–107.

[10]

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.

[11]

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.

[12]

G. Cerami and D. Passaseo, Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with "rich" topology, Nonlinear Anal. TMA, 18 (1992), 109-119.  doi: 10.1016/0362-546X(92)90089-W.

[13]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[14]

S. Cingolani, L. Jeanjean and K. Tanaka, Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439.  doi: 10.1007/s00526-014-0754-5.

[15]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differ. Equ., 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.

[16]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem, Top. Methods Nonlinear Anal., 4 (1999), 347-368.  doi: 10.12775/TMNA.1998.016.

[17]

E. N. Dancer and S. Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Differ. Equ., 4 (1999), 347–368.

[18]

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

[19]

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.

[20]

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

[21]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 15. No. 2. Elsevier Masson, 1998, 127–149. doi: 10.1016/S0294-1449(97)89296-7.

[22]

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.

[23]

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

[24]

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

[25]

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.

[26]

Y. H. 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.

[27]

Y. H. 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.

[28]

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

[29]

J. Esteban and Eric Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 323–350.

[30]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields, Physical Review, 83 (1951), 326–332.

[31]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field, Physical Review, 103 (1956), 1571–1579.

[32]

G. Fournier and M. Willem, Relative category and the calculus of variations, in Variational Methods, H. Berestycki et al. Birkhäuser Boston, 4 (1990), 95–104.

[33]

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.

[34]

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

[35]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, Part Ⅱ, AIP Anal. non linéaire, 1, 223–283.

[36]

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.

[37]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223–253.

[38]

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

[39]

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.

[40]

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.

[41]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

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