May  2015, 14(3): 1097-1125. doi: 10.3934/cpaa.2015.14.1097

Klein-Gordon-Maxwell equations in high dimensions

1. 

Université de Cergy-Pontoise, CNRS, Département de Mathématiques, F-95000 Cergy-Pontoise, France

Published  March 2015

We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimensions.
Citation: Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097
References:
[1]

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  Google Scholar

[2]

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

[3]

A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449-463. doi: 10.1017/S0308210509001814.  Google Scholar

[4]

Antonio Azzollini and Alessio Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.  Google Scholar

[5]

Vieri Benci and Claudio Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter, Adv. Nonlinear Stud., 12 (2012), 717-735.  Google Scholar

[6]

Vieri Benci and Donato Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.  Google Scholar

[7]

Vieri Benci and Donato Fortunato, Solitary waves in the nonlinear wave equation and in gauge theories, J. Fixed Point Theory Appl., 1 (2007), 61-86. doi: 10.1007/s11784-006-0008-z.  Google Scholar

[8]

Vieri Benci and Donato Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 20 (2009), 243-279. doi: 10.4171/RLM/546.  Google Scholar

[9]

Vieri Benci and Donato Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Comm. Math. Phys., 295 (2010), 639-668. doi: 10.1007/s00220-010-0985-z.  Google Scholar

[10]

Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[11]

Luis A. Caffarelli, Basilis Gidas and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[12]

Daniele Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747. doi: 10.1016/j.na.2003.05.001.  Google Scholar

[13]

Monica Clapp, Marco Ghimenti and Anna Maria Micheletti, Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold, Preprint, (2013). Google Scholar

[14]

Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 23 (2004), 1143-1191. doi: 10.1155/S1073792804133278.  Google Scholar

[15]

Olivier Druet and Emmanuel Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE, 2 (2009), 305-359. doi: 10.2140/apde.2009.2.305.  Google Scholar

[16]

Olivier Druet and Emmanuel Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831-869. doi: 10.1142/S0219199710004007.  Google Scholar

[17]

Olivier Druet, Emmanuel Hebey and Paul Laurain, Stability of elliptic PDEs with respect to perturbations of the domain, J. Differential Equations, 255 (2013), 3703-3718. doi: 10.1016/j.jde.2013.07.051.  Google Scholar

[18]

Olivier Druet, Emmanuel Hebey and Frédéric Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, 45, Princeton University Press, Princeton, NJ, 2004. doi: 10.1007/BF01158557.  Google Scholar

[19]

Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059. doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[20]

Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Stable phases for the 4-dimensional KGMP system.,, \emph{J. Reine Angew. Math.}, ().   Google Scholar

[21]

Olivier Druet and Paul Laurain, Stability of the Pohožaev obstruction in dimension 3, J. Eur. Math. Soc. (JEMS), 12 (2010), 1117-1149. doi: 10.4171/JEMS/225.  Google Scholar

[22]

Teresa D'Aprile and Dimitri Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  Google Scholar

[23]

Teresa D'Aprile and Dimitri Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.  Google Scholar

[24]

Teresa D'Aprile and Juncheng Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. doi: 10.1016/j.jde.2005.12.009.  Google Scholar

[25]

Teresa D'Aprile and Juncheng Wei, Solutions en grappe autour des centres harmoniques d'un système elliptique couplé, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 605-628. doi: 10.1016/j.anihpc.2006.04.003.  Google Scholar

[26]

Pietro d'Avenia and Lorenzo Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations, Electron. J. Differential Equations, (2002).  Google Scholar

[27]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.02.111.  Google Scholar

[28]

Pietro d'Avenia, Lorenzo Pisani and Gaetano Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149. doi: 10.3934/dcds.2010.26.135.  Google Scholar

[29]

Pierpaolo Esposito, Angela Pistoia and Jérôme Vétois, The effect of linear perturbations on the Yamabe problem,, \emph{Math. Ann.}, ().  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[30]

Vladimir Georgiev and Nicola Visciglia, Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential, J. Math. Pures Appl., 84 (2005), 957-983. doi: 10.1016/j.matpur.2004.09.016.  Google Scholar

[31]

Marco Ghimenti and Anna Maria Micheletti, Number and profile of low energy solutions for singularly perturbed Klein-Gordon-Maxwell systems on a Riemannian manifold, Preprint, (2013). doi: 10.1016/j.jde.2014.01.012.  Google Scholar

[32]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[33]

David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[34]

Emmanuel Hebey, Solitary waves in critical abelian gauge theories, Discrete Contin. Dyn. Syst., 32 (2012), 1747-1761. doi: 10.3934/dcds.2012.32.1747.  Google Scholar

[35]

Emmanuel Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, European Mathematical Society, ().  doi: 10.4171/134.  Google Scholar

[36]

Emmanuel Hebey and Juncheng Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967. doi: 10.4310/MRL.2012.v19.n4.a18.  Google Scholar

[37]

Emmanuel Hebey and Juncheng Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0.  Google Scholar

[38]

Emmanuel Hebey and Trong Tuong Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds, J. Reine Angew. Math., 667 (2012), 221-248.  Google Scholar

[39]

Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279. doi: 10.1215/S0012-7094-95-07906-X.  Google Scholar

[40]

YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[41]

YanYan Li and Lei Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151. doi: 10.1007/s00526-003-0224-y.  Google Scholar

[42]

YanYan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations, 24 (2005), 185-237. doi: 10.1007/s00526-004-0320-7.  Google Scholar

[43]

Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50. doi: 10.1142/S021919979900002X.  Google Scholar

[44]

Dimitri Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1519-1527. doi: 10.1098/rspa.2003.1267.  Google Scholar

[45]

Dimitri Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1055-1071. doi: 10.1016/j.anihpc.2010.02.001.  Google Scholar

[46]

Frédérique Robert and Jérôme Vétois, Examples of non-isolated blow-up for perturbations of the scalar curvature equation, Preprint, (2012). Google Scholar

[47]

Frédérique Robert and Jérôme Vétois, A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations with Lyapunov-Schmidt's Finite-dimensional Reduction, Cocompact Imbeddings, Profile Decompositions, and their Applications to PDE, Trends Math., Birkhäuser, 2013, 85-116. Google Scholar

[48]

Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.  Google Scholar

[49]

Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.  Google Scholar

show all references

References:
[1]

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  Google Scholar

[2]

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

[3]

A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449-463. doi: 10.1017/S0308210509001814.  Google Scholar

[4]

Antonio Azzollini and Alessio Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.  Google Scholar

[5]

Vieri Benci and Claudio Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter, Adv. Nonlinear Stud., 12 (2012), 717-735.  Google Scholar

[6]

Vieri Benci and Donato Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.  Google Scholar

[7]

Vieri Benci and Donato Fortunato, Solitary waves in the nonlinear wave equation and in gauge theories, J. Fixed Point Theory Appl., 1 (2007), 61-86. doi: 10.1007/s11784-006-0008-z.  Google Scholar

[8]

Vieri Benci and Donato Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 20 (2009), 243-279. doi: 10.4171/RLM/546.  Google Scholar

[9]

Vieri Benci and Donato Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Comm. Math. Phys., 295 (2010), 639-668. doi: 10.1007/s00220-010-0985-z.  Google Scholar

[10]

Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[11]

Luis A. Caffarelli, Basilis Gidas and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[12]

Daniele Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747. doi: 10.1016/j.na.2003.05.001.  Google Scholar

[13]

Monica Clapp, Marco Ghimenti and Anna Maria Micheletti, Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold, Preprint, (2013). Google Scholar

[14]

Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 23 (2004), 1143-1191. doi: 10.1155/S1073792804133278.  Google Scholar

[15]

Olivier Druet and Emmanuel Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE, 2 (2009), 305-359. doi: 10.2140/apde.2009.2.305.  Google Scholar

[16]

Olivier Druet and Emmanuel Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831-869. doi: 10.1142/S0219199710004007.  Google Scholar

[17]

Olivier Druet, Emmanuel Hebey and Paul Laurain, Stability of elliptic PDEs with respect to perturbations of the domain, J. Differential Equations, 255 (2013), 3703-3718. doi: 10.1016/j.jde.2013.07.051.  Google Scholar

[18]

Olivier Druet, Emmanuel Hebey and Frédéric Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, 45, Princeton University Press, Princeton, NJ, 2004. doi: 10.1007/BF01158557.  Google Scholar

[19]

Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059. doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[20]

Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Stable phases for the 4-dimensional KGMP system.,, \emph{J. Reine Angew. Math.}, ().   Google Scholar

[21]

Olivier Druet and Paul Laurain, Stability of the Pohožaev obstruction in dimension 3, J. Eur. Math. Soc. (JEMS), 12 (2010), 1117-1149. doi: 10.4171/JEMS/225.  Google Scholar

[22]

Teresa D'Aprile and Dimitri Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  Google Scholar

[23]

Teresa D'Aprile and Dimitri Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.  Google Scholar

[24]

Teresa D'Aprile and Juncheng Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. doi: 10.1016/j.jde.2005.12.009.  Google Scholar

[25]

Teresa D'Aprile and Juncheng Wei, Solutions en grappe autour des centres harmoniques d'un système elliptique couplé, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 605-628. doi: 10.1016/j.anihpc.2006.04.003.  Google Scholar

[26]

Pietro d'Avenia and Lorenzo Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations, Electron. J. Differential Equations, (2002).  Google Scholar

[27]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.02.111.  Google Scholar

[28]

Pietro d'Avenia, Lorenzo Pisani and Gaetano Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149. doi: 10.3934/dcds.2010.26.135.  Google Scholar

[29]

Pierpaolo Esposito, Angela Pistoia and Jérôme Vétois, The effect of linear perturbations on the Yamabe problem,, \emph{Math. Ann.}, ().  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[30]

Vladimir Georgiev and Nicola Visciglia, Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential, J. Math. Pures Appl., 84 (2005), 957-983. doi: 10.1016/j.matpur.2004.09.016.  Google Scholar

[31]

Marco Ghimenti and Anna Maria Micheletti, Number and profile of low energy solutions for singularly perturbed Klein-Gordon-Maxwell systems on a Riemannian manifold, Preprint, (2013). doi: 10.1016/j.jde.2014.01.012.  Google Scholar

[32]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[33]

David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[34]

Emmanuel Hebey, Solitary waves in critical abelian gauge theories, Discrete Contin. Dyn. Syst., 32 (2012), 1747-1761. doi: 10.3934/dcds.2012.32.1747.  Google Scholar

[35]

Emmanuel Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, European Mathematical Society, ().  doi: 10.4171/134.  Google Scholar

[36]

Emmanuel Hebey and Juncheng Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967. doi: 10.4310/MRL.2012.v19.n4.a18.  Google Scholar

[37]

Emmanuel Hebey and Juncheng Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0.  Google Scholar

[38]

Emmanuel Hebey and Trong Tuong Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds, J. Reine Angew. Math., 667 (2012), 221-248.  Google Scholar

[39]

Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279. doi: 10.1215/S0012-7094-95-07906-X.  Google Scholar

[40]

YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[41]

YanYan Li and Lei Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151. doi: 10.1007/s00526-003-0224-y.  Google Scholar

[42]

YanYan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations, 24 (2005), 185-237. doi: 10.1007/s00526-004-0320-7.  Google Scholar

[43]

Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50. doi: 10.1142/S021919979900002X.  Google Scholar

[44]

Dimitri Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1519-1527. doi: 10.1098/rspa.2003.1267.  Google Scholar

[45]

Dimitri Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1055-1071. doi: 10.1016/j.anihpc.2010.02.001.  Google Scholar

[46]

Frédérique Robert and Jérôme Vétois, Examples of non-isolated blow-up for perturbations of the scalar curvature equation, Preprint, (2012). Google Scholar

[47]

Frédérique Robert and Jérôme Vétois, A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations with Lyapunov-Schmidt's Finite-dimensional Reduction, Cocompact Imbeddings, Profile Decompositions, and their Applications to PDE, Trends Math., Birkhäuser, 2013, 85-116. Google Scholar

[48]

Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.  Google Scholar

[49]

Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.  Google Scholar

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Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066

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