• Previous Article
    An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue
  • CPAA Home
  • This Issue
  • Next Article
    On existence and nonexistence of the positive solutions of non-newtonian filtration equation
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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

M. Willem, "Minimax Theorems,", Birkhuser Boston, (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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[1]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135

[2]

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

[3]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271

[4]

Sitong Chen, Xianhua Tang. Improved results for Klein-Gordon-Maxwell systems with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2333-2348. doi: 10.3934/dcds.2018096

[5]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[6]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669

[7]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573

[8]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034

[9]

Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139

[10]

Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221

[11]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Global attractor for a Klein-Gordon-Schrodinger type system. Conference Publications, 2007, 2007 (Special) : 844-854. doi: 10.3934/proc.2007.2007.844

[12]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233

[13]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149

[14]

Julián López-Gómez. Uniqueness of radially symmetric large solutions. Conference Publications, 2007, 2007 (Special) : 677-686. doi: 10.3934/proc.2007.2007.677

[15]

Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054

[16]

Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427

[17]

Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753

[18]

Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030

[19]

Mónica Clapp, Marco Squassina. Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Communications on Pure & Applied Analysis, 2003, 2 (2) : 171-186. doi: 10.3934/cpaa.2003.2.171

[20]

Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

[Back to Top]