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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, 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] | |
[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] | |
[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.
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