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March  2011, 10(2): 709-718. doi: 10.3934/cpaa.2011.10.709

Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents

1. 

Departamento de Matematica, Universidade Federal de Minas Gerais, 31270-010 Belo Horizonte-MG, Brazil

2. 

Departamento Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil

3. 

Departamento de Matematica, Universidade Federal de Vicosa, 36371-000 Vicosa-MG

Received  May 2010 Revised  August 2010 Published  December 2010

In this paper we study the existence of radially symmetric solitary waves in $R^N$ for the nonlinear Klein-Gordon equations coupled with the Maxwell's equations when the nonlinearity exhibits critical growth. The main feature of this kind of problem is the lack of compactness arising in connection with the use of variational methods.
Citation: Paulo Cesar Carrião, Patrícia L. Cunha, Olímpio Hiroshi Miyagaki. Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Communications on Pure & Applied Analysis, 2011, 10 (2) : 709-718. doi: 10.3934/cpaa.2011.10.709
References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and aplications,, J. Functional Analysis, 14 (1973), 349. doi: doi:10.1016/0022-1236(73)90051-7. Google Scholar

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. Google Scholar

[3]

H. Berestycki and P. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: doi:10.1007/BF00250556. Google Scholar

[4]

V. Benci and D. Fortunato, The nonlinear Klein-Gordon equation coupled with the Maxwell equations,, Nonlinear Anal., 47 (2001), 6065. doi: doi:10.1016/S0362-546X(01)00688-5. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schröinger-Maxwell equations,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893. doi: doi:10.1017/S030821050000353X. Google Scholar

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar

[10]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009), 1985. doi: doi:10.1016/j.na.2009.02.111. Google Scholar

[11]

P. d'Avenia, L. Pisani, and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. Google Scholar

[12]

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

[13]

D. 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. Google Scholar

[14]

O. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: doi:10.1016/S0362-546X(96)00087-9. Google Scholar

[15]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: doi:10.1007/BF02418013. Google Scholar

[16]

M. Willem, "Minimax Theorems,", Birkhuser Boston, (1996). Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and aplications,, J. Functional Analysis, 14 (1973), 349. doi: doi:10.1016/0022-1236(73)90051-7. Google Scholar

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. Google Scholar

[3]

H. Berestycki and P. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: doi:10.1007/BF00250556. Google Scholar

[4]

V. Benci and D. Fortunato, The nonlinear Klein-Gordon equation coupled with the Maxwell equations,, Nonlinear Anal., 47 (2001), 6065. doi: doi:10.1016/S0362-546X(01)00688-5. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schröinger-Maxwell equations,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893. doi: doi:10.1017/S030821050000353X. Google Scholar

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar

[10]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009), 1985. doi: doi:10.1016/j.na.2009.02.111. Google Scholar

[11]

P. d'Avenia, L. Pisani, and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. Google Scholar

[12]

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

[13]

D. 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. Google Scholar

[14]

O. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: doi:10.1016/S0362-546X(96)00087-9. Google Scholar

[15]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: doi:10.1007/BF02418013. Google Scholar

[16]

M. Willem, "Minimax Theorems,", Birkhuser Boston, (1996). Google Scholar

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