June  2012, 32(6): 2271-2283. doi: 10.3934/dcds.2012.32.2271

Nonradial solutions for the Klein-Gordon-Maxwell equations

1. 

Mathematical Institute, University of Giessen, Arndtstr. 2, D–35392 Giessen, Germany

Received  April 2011 Revised  October 2011 Published  February 2012

We study a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations. We prove the existence of nonradial solutions which are radially symmetric when restricted to a hyperplane, and either periodic or non-periodic in the orthogonal direction to that very hyperplane.
Citation: Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271
References:
[1]

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

[2]

A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763. doi: 10.1016/j.jde.2010.07.007.  Google Scholar

[3]

M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. doi: 10.4171/JEMS/83.  Google Scholar

[4]

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: 10.1142/S0129055X02001168.  Google Scholar

[5]

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: 10.1016/j.na.2003.05.001.  Google Scholar

[6]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  Google Scholar

[7]

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

[8]

P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.  Google Scholar

[9]

M. J. Esteban and P.-L. Lions, A compactness lemma, Nonlinear Analysis, 7 (1983), 381-385.  Google Scholar

[10]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  Google Scholar

[11]

P.-L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[12]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[13]

M. Willem, "Minimax Theorems,'' Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, 1996.  Google Scholar

show all references

References:
[1]

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

[2]

A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763. doi: 10.1016/j.jde.2010.07.007.  Google Scholar

[3]

M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. doi: 10.4171/JEMS/83.  Google Scholar

[4]

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: 10.1142/S0129055X02001168.  Google Scholar

[5]

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: 10.1016/j.na.2003.05.001.  Google Scholar

[6]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  Google Scholar

[7]

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

[8]

P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.  Google Scholar

[9]

M. J. Esteban and P.-L. Lions, A compactness lemma, Nonlinear Analysis, 7 (1983), 381-385.  Google Scholar

[10]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  Google Scholar

[11]

P.-L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[12]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'' CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[13]

M. Willem, "Minimax Theorems,'' Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, 1996.  Google Scholar

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