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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

Paulo Cesar Carrião 1, , Patrícia L. Cunha 2, and  Olímpio Hiroshi Miyagaki 3,

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.
Keywords: critical growth., Klein-Gordon-Maxwell system, radially symmetric solution.
Mathematics Subject Classification: Primary: 35J47, 35J50, 35B33; Secondary: 81Q05, 34B1.
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 and 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), 349-381. doi: doi:10.1016/0022-1236(73)90051-7.

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[3]

H. Berestycki and P. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: doi:10.1007/BF00250556.

[4]

V. Benci and D. Fortunato, The nonlinear Klein-Gordon equation coupled with the Maxwell equations, Nonlinear Anal., 47 (2001), 6065-6072. doi: doi:10.1016/S0362-546X(01)00688-5.

[5]

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

[6]

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

[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), 733-747. doi: doi:10.1016/j.na.2003.05.001.

[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), 893-906. doi: doi:10.1017/S030821050000353X.

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.

[10]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), 1985-1995. doi: doi:10.1016/j.na.2009.02.111.

[11]

P. d'Avenia, L. Pisani, and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149.

[12]

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

[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), 1519-1527.

[14]

O. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781. doi: doi:10.1016/S0362-546X(96)00087-9.

[15]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: doi:10.1007/BF02418013.

[16]

M. Willem, "Minimax Theorems," Birkh鋟ser Boston, Inc., Boston, MA, 1996.

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and aplications, J. Functional Analysis, 14 (1973), 349-381. doi: doi:10.1016/0022-1236(73)90051-7.

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[3]

H. Berestycki and P. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: doi:10.1007/BF00250556.

[4]

V. Benci and D. Fortunato, The nonlinear Klein-Gordon equation coupled with the Maxwell equations, Nonlinear Anal., 47 (2001), 6065-6072. doi: doi:10.1016/S0362-546X(01)00688-5.

[5]

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

[6]

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

[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), 733-747. doi: doi:10.1016/j.na.2003.05.001.

[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), 893-906. doi: doi:10.1017/S030821050000353X.

[9]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.

[10]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), 1985-1995. doi: doi:10.1016/j.na.2009.02.111.

[11]

P. d'Avenia, L. Pisani, and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149.

[12]

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

[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), 1519-1527.

[14]

O. Miyagaki, On a class of semilinear elliptic problems in $R^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781. doi: doi:10.1016/S0362-546X(96)00087-9.

[15]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: doi:10.1007/BF02418013.

[16]

M. Willem, "Minimax Theorems," Birkh鋟ser Boston, Inc., Boston, MA, 1996.

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