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

On Dirac equation with a potential and critical Sobolev exponent

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.
Citation: Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
References:
[1]

R. Adams, Sobolev Space, $2^{nd}$ edition, Academic Press, New York 1975, 208-221.

[2]

S. Alama and Yanyan Li, Existence of solutions for nonlinear elliptic euqations with indefinite linear part, J. Diff. Equa., 96 (1922), 85-115. doi: 10.1016/0022-0396(92)90145-D.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[4]

B. Ammann, A variational Problem in Conformal Spin Geometry, Ph.D thesis, Habilitationsschift, Universität Hamburg, 2003.

[5]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Comm. Anal. Geom., 17 (2009), 429-479. doi: 10.4310/CAG.2009.v17.n3.a2.

[6]

B. Ammann, J.-F. Grosjean, E. Humbert and B. Morel, A spinorial analogue of Aubin's inequality, Math.Z., 260 (2008), 127-151. doi: 10.1007/s00209-007-0266-5.

[7]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296.

[8]

A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Commun. Pure Appl. Math., 37 (1984), 403-432. doi: 10.1002/cpa.3160370402.

[9]

A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system, Acta Math., 152 (1984), 143-197. doi: 10.1007/BF02392196.

[10]

C. Bär, Metrics with harmonic spinors, Geom. Funct. Anal., 6 (1996), 899-942. doi: 10.1007/BF02246994.

[11]

H. Brezis, J. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-689. doi: 10.1002/cpa.3160330507.

[12]

V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun.Pure Appl. Math., 31 (1978), 157-184.

[13]

J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques, Comm. Math. Phys., 144 (1992), 581-599.

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 431-477. doi: 10.1002/cpa.3160360405.

[15]

T. Friedrich, Dirac Operators in Riemannian Geometry, Grad. Stud. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/gsm/025.

[16]

N. Ginoux, The Dirac Spectrum, Lecture Notes in Math.,vol. 1976, Springer, Dordrecht-heidelberg-London-New York, 2009. doi: 10.1007/978-3-642-01570-0.

[17]

O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys., 104 (1986), 151-162.

[18]

N. Hitchin, Harmonic spinors, Adv. Math., 14 (1974), 1-55.

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, Manuscripta math., 135 (2011), 329-360. doi: 10.1007/s00229-010-0417-6.

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds, J. Funct. Anal., 260 (2011), 253-307. doi: 10.1016/j.jfa.2010.09.008.

[21]

C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent, Topol. Meth. Nonl. Anal., 12 (1988), 245-261.

[22]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.

[23]

H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989.

[24]

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

[25]

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

[26]

S. Rault, A Sobolev-like inequality for the Dirac operator, J. Funct. Anal., 26 (2009), 1588-1617. doi: 10.1016/j.jfa.2008.11.007.

[27]

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

[28]

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

show all references

References:
[1]

R. Adams, Sobolev Space, $2^{nd}$ edition, Academic Press, New York 1975, 208-221.

[2]

S. Alama and Yanyan Li, Existence of solutions for nonlinear elliptic euqations with indefinite linear part, J. Diff. Equa., 96 (1922), 85-115. doi: 10.1016/0022-0396(92)90145-D.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[4]

B. Ammann, A variational Problem in Conformal Spin Geometry, Ph.D thesis, Habilitationsschift, Universität Hamburg, 2003.

[5]

B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Comm. Anal. Geom., 17 (2009), 429-479. doi: 10.4310/CAG.2009.v17.n3.a2.

[6]

B. Ammann, J.-F. Grosjean, E. Humbert and B. Morel, A spinorial analogue of Aubin's inequality, Math.Z., 260 (2008), 127-151. doi: 10.1007/s00209-007-0266-5.

[7]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296.

[8]

A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Commun. Pure Appl. Math., 37 (1984), 403-432. doi: 10.1002/cpa.3160370402.

[9]

A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system, Acta Math., 152 (1984), 143-197. doi: 10.1007/BF02392196.

[10]

C. Bär, Metrics with harmonic spinors, Geom. Funct. Anal., 6 (1996), 899-942. doi: 10.1007/BF02246994.

[11]

H. Brezis, J. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-689. doi: 10.1002/cpa.3160330507.

[12]

V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun.Pure Appl. Math., 31 (1978), 157-184.

[13]

J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques, Comm. Math. Phys., 144 (1992), 581-599.

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 431-477. doi: 10.1002/cpa.3160360405.

[15]

T. Friedrich, Dirac Operators in Riemannian Geometry, Grad. Stud. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/gsm/025.

[16]

N. Ginoux, The Dirac Spectrum, Lecture Notes in Math.,vol. 1976, Springer, Dordrecht-heidelberg-London-New York, 2009. doi: 10.1007/978-3-642-01570-0.

[17]

O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys., 104 (1986), 151-162.

[18]

N. Hitchin, Harmonic spinors, Adv. Math., 14 (1974), 1-55.

[19]

T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, Manuscripta math., 135 (2011), 329-360. doi: 10.1007/s00229-010-0417-6.

[20]

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds, J. Funct. Anal., 260 (2011), 253-307. doi: 10.1016/j.jfa.2010.09.008.

[21]

C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent, Topol. Meth. Nonl. Anal., 12 (1988), 245-261.

[22]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.

[23]

H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton University Press, 1989.

[24]

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

[25]

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

[26]

S. Rault, A Sobolev-like inequality for the Dirac operator, J. Funct. Anal., 26 (2009), 1588-1617. doi: 10.1016/j.jfa.2008.11.007.

[27]

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

[28]

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

[1]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[2]

Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2002, 8 (2) : 381-397. doi: 10.3934/dcds.2002.8.381

[13]

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

[14]

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

[15]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and 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 and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[17]

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

[18]

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

[19]

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

[20]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (164)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]