American Institute of Mathematical Sciences

Advanced
November  2015, 14(6): 2231-2263. doi: 10.3934/cpaa.2015.14.2231

On Dirac equation with a potential and critical Sobolev exponent

Wenmin Gong 1, and  Guangcun Lu 2,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received  November 2014 Revised  May 2015 Published  September 2015

In this paper we consider a critical Dirac equation with a potential on a compact spin manifold. We prove the existence of solutions based on the analysis of the spectrum of Dirac operator with a potential and the dual variational method.
Keywords: Strongly indefinite functionals., Nonlinear Dirac equation, Variational methods.
Mathematics Subject Classification: Primary: 58J05, 58E05, 53C2.
Citation: Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
References:
[1]

R. Adams, Sobolev Space,, $2^{nd}$ edition, (1975), 208.   Google Scholar

[2]

S. Alama and Yanyan Li, Existence of solutions for nonlinear elliptic euqations with indefinite linear part,, \emph{J. Diff. Equa.}, 96 (1922), 85.  doi: 10.1016/0022-0396(92)90145-D.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.   Google Scholar

[4]

B. Ammann, A variational Problem in Conformal Spin Geometry,, Ph.D thesis, (2003).   Google Scholar

[5]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions,, \emph{Comm. Anal. Geom.}, 17 (2009), 429.  doi: 10.4310/CAG.2009.v17.n3.a2.  Google Scholar

[6]

B. Ammann, J.-F. Grosjean, E. Humbert and B. Morel, A spinorial analogue of Aubin's inequality,, \emph{Math.Z.}, 260 (2008), 127.  doi: 10.1007/s00209-007-0266-5.  Google Scholar

[7]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269.   Google Scholar

[8]

A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations,, \emph{Commun. Pure Appl. Math.}, 37 (1984), 403.  doi: 10.1002/cpa.3160370402.  Google Scholar

[9]

A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system,, \emph{Acta Math.}, 152 (1984), 143.  doi: 10.1007/BF02392196.  Google Scholar

[10]

C. Bär, Metrics with harmonic spinors,, \emph{Geom. Funct. Anal.}, 6 (1996), 899.  doi: 10.1007/BF02246994.  Google Scholar

[11]

H. Brezis, J. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,, \emph{Comm. Pure Appl. Math.}, 33 (1980), 667.  doi: 10.1002/cpa.3160330507.  Google Scholar

[12]

V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, \emph{Commun.Pure Appl. Math.}, 31 (1978), 157.   Google Scholar

[13]

J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques,, \emph{Comm. Math. Phys.}, 144 (1992), 581.   Google Scholar

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.} \textbf{36} (1983), 36 (1983), 431.  doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

T. Friedrich, Dirac Operators in Riemannian Geometry,, Grad. Stud. Math., (2000).  doi: 10.1090/gsm/025.  Google Scholar

[16]

N. Ginoux, The Dirac Spectrum,, Lecture Notes in Math., (1976).  doi: 10.1007/978-3-642-01570-0.  Google Scholar

[17]

O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors,, \emph{Comm. Math. Phys.}, 104 (1986), 151.   Google Scholar

[18]

N. Hitchin, Harmonic spinors,, \emph{Adv. Math.}, 14 (1974), 1.   Google Scholar

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds,, \emph{Manuscripta math.}, 135 (2011), 329.  doi: 10.1007/s00229-010-0417-6.  Google Scholar

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds,, \emph{J. Funct. Anal.}, 260 (2011), 253.  doi: 10.1016/j.jfa.2010.09.008.  Google Scholar

[21]

C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent,, \emph{Topol. Meth. Nonl. Anal.}, 12 (1988), 245.   Google Scholar

[22]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation,, \emph{Adv. Differential Equations}, 3 (1998), 441.   Google Scholar

[23]

H. B. Lawson and M. L. Michelson, Spin Geometry,, Princeton University Press, (1989).   Google Scholar

[24]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I,, 1972., ().   Google Scholar

[25]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations., In:CBMS Reg. Conf. Ser. No. 65 AMS, (1986).   Google Scholar

[26]

S. Rault, A Sobolev-like inequality for the Dirac operator,, \emph{J. Funct. Anal.}, 26 (2009), 1588.  doi: 10.1016/j.jfa.2008.11.007.  Google Scholar

[27]

M. Reed and B. Simon, Methods of Mathematical Physics, vols. I-IV ,, Academic Press, (1978).   Google Scholar

[28]

J. Wolf, Essential self-adjointness for the Dirac operator and its square,, \emph{Indiana Univ. Math. J.}, 22 (): 611.   Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Space,, $2^{nd}$ edition, (1975), 208.   Google Scholar

[2]

S. Alama and Yanyan Li, Existence of solutions for nonlinear elliptic euqations with indefinite linear part,, \emph{J. Diff. Equa.}, 96 (1922), 85.  doi: 10.1016/0022-0396(92)90145-D.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.   Google Scholar

[4]

B. Ammann, A variational Problem in Conformal Spin Geometry,, Ph.D thesis, (2003).   Google Scholar

[5]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions,, \emph{Comm. Anal. Geom.}, 17 (2009), 429.  doi: 10.4310/CAG.2009.v17.n3.a2.  Google Scholar

[6]

B. Ammann, J.-F. Grosjean, E. Humbert and B. Morel, A spinorial analogue of Aubin's inequality,, \emph{Math.Z.}, 260 (2008), 127.  doi: 10.1007/s00209-007-0266-5.  Google Scholar

[7]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269.   Google Scholar

[8]

A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations,, \emph{Commun. Pure Appl. Math.}, 37 (1984), 403.  doi: 10.1002/cpa.3160370402.  Google Scholar

[9]

A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system,, \emph{Acta Math.}, 152 (1984), 143.  doi: 10.1007/BF02392196.  Google Scholar

[10]

C. Bär, Metrics with harmonic spinors,, \emph{Geom. Funct. Anal.}, 6 (1996), 899.  doi: 10.1007/BF02246994.  Google Scholar

[11]

H. Brezis, J. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,, \emph{Comm. Pure Appl. Math.}, 33 (1980), 667.  doi: 10.1002/cpa.3160330507.  Google Scholar

[12]

V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, \emph{Commun.Pure Appl. Math.}, 31 (1978), 157.   Google Scholar

[13]

J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques,, \emph{Comm. Math. Phys.}, 144 (1992), 581.   Google Scholar

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.} \textbf{36} (1983), 36 (1983), 431.  doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

T. Friedrich, Dirac Operators in Riemannian Geometry,, Grad. Stud. Math., (2000).  doi: 10.1090/gsm/025.  Google Scholar

[16]

N. Ginoux, The Dirac Spectrum,, Lecture Notes in Math., (1976).  doi: 10.1007/978-3-642-01570-0.  Google Scholar

[17]

O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors,, \emph{Comm. Math. Phys.}, 104 (1986), 151.   Google Scholar

[18]

N. Hitchin, Harmonic spinors,, \emph{Adv. Math.}, 14 (1974), 1.   Google Scholar

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds,, \emph{Manuscripta math.}, 135 (2011), 329.  doi: 10.1007/s00229-010-0417-6.  Google Scholar

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds,, \emph{J. Funct. Anal.}, 260 (2011), 253.  doi: 10.1016/j.jfa.2010.09.008.  Google Scholar

[21]

C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent,, \emph{Topol. Meth. Nonl. Anal.}, 12 (1988), 245.   Google Scholar

[22]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation,, \emph{Adv. Differential Equations}, 3 (1998), 441.   Google Scholar

[23]

H. B. Lawson and M. L. Michelson, Spin Geometry,, Princeton University Press, (1989).   Google Scholar

[24]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I,, 1972., ().   Google Scholar

[25]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations., In:CBMS Reg. Conf. Ser. No. 65 AMS, (1986).   Google Scholar

[26]

S. Rault, A Sobolev-like inequality for the Dirac operator,, \emph{J. Funct. Anal.}, 26 (2009), 1588.  doi: 10.1016/j.jfa.2008.11.007.  Google Scholar

[27]

M. Reed and B. Simon, Methods of Mathematical Physics, vols. I-IV ,, Academic Press, (1978).   Google Scholar

[28]

J. Wolf, Essential self-adjointness for the Dirac operator and its square,, \emph{Indiana Univ. Math. J.}, 22 (): 611.   Google Scholar

[1]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020297
[2]

Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723
[3]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[4]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227
[5]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[6]

Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623
[7]

Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
[8]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189
[9]

Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[10]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[11]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[12]

Maria J. Esteban, Eric Séré. An overview on linear and nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 381-397. doi: 10.3934/dcds.2002.8.381
[13]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[14]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[15]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[16]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[17]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[18]

Philippe Michel, Bhargav Kumar Kakumani. GRE methods for nonlinear model of evolution equation and limited ressource environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6653-6673. doi: 10.3934/dcdsb.2019161
[19]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, , () : -. doi: 10.3934/era.2020079
[20]

Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences