American Institute of Mathematical Sciences

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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

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