# American Institute of Mathematical Sciences

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 & 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. [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. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. [4] B. Ammann, A variational Problem in Conformal Spin Geometry,, Ph.D thesis, (2003). [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. [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. [7] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269. [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. [9] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system,, \emph{Acta Math.}, 152 (1984), 143. doi: 10.1007/BF02392196. [10] C. Bär, Metrics with harmonic spinors,, \emph{Geom. Funct. Anal.}, 6 (1996), 899. 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,, \emph{Comm. Pure Appl. Math.}, 33 (1980), 667. doi: 10.1002/cpa.3160330507. [12] V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, \emph{Commun.Pure Appl. Math.}, 31 (1978), 157. [13] J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques,, \emph{Comm. Math. Phys.}, 144 (1992), 581. [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. [15] T. Friedrich, Dirac Operators in Riemannian Geometry,, Grad. Stud. Math., (2000). doi: 10.1090/gsm/025. [16] N. Ginoux, The Dirac Spectrum,, Lecture Notes in Math., (1976). 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,, \emph{Comm. Math. Phys.}, 104 (1986), 151. [18] N. Hitchin, Harmonic spinors,, \emph{Adv. Math.}, 14 (1974), 1. [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. [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. [21] C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent,, \emph{Topol. Meth. Nonl. Anal.}, 12 (1988), 245. [22] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation,, \emph{Adv. Differential Equations}, 3 (1998), 441. [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, (1986). [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. [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.

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##### References:
 [1] R. Adams, Sobolev Space,, $2^{nd}$ edition, (1975), 208. [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. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. [4] B. Ammann, A variational Problem in Conformal Spin Geometry,, Ph.D thesis, (2003). [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. [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. [7] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269. [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. [9] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian system,, \emph{Acta Math.}, 152 (1984), 143. doi: 10.1007/BF02392196. [10] C. Bär, Metrics with harmonic spinors,, \emph{Geom. Funct. Anal.}, 6 (1996), 899. 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,, \emph{Comm. Pure Appl. Math.}, 33 (1980), 667. doi: 10.1002/cpa.3160330507. [12] V. Benci and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, \emph{Commun.Pure Appl. Math.}, 31 (1978), 157. [13] J. P. Bourguignon and P. Gauduchon, Spineurs, Opérateurs de Dirac et variations de métriques,, \emph{Comm. Math. Phys.}, 144 (1992), 581. [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. [15] T. Friedrich, Dirac Operators in Riemannian Geometry,, Grad. Stud. Math., (2000). doi: 10.1090/gsm/025. [16] N. Ginoux, The Dirac Spectrum,, Lecture Notes in Math., (1976). 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,, \emph{Comm. Math. Phys.}, 104 (1986), 151. [18] N. Hitchin, Harmonic spinors,, \emph{Adv. Math.}, 14 (1974), 1. [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. [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. [21] C. Jan and Y. Jianfu, On Schrödinger equation with periodic potential and critical Sobolev exponent,, \emph{Topol. Meth. Nonl. Anal.}, 12 (1988), 245. [22] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation,, \emph{Adv. Differential Equations}, 3 (1998), 441. [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, (1986). [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. [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.
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