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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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