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An improved result for the full justification of asymptotic models for the propagation of internal waves
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 |
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). 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. |
[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). Google Scholar |
[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, (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). 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. |
[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). Google Scholar |
[28] |
J. Wolf, Essential self-adjointness for the Dirac operator and its square,, \emph{Indiana Univ. Math. J.}, 22 (): 611.
|
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