# American Institute of Mathematical Sciences

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 - A, 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. doi: 10.1016/0022-1236(73)90051-7. [2] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity,, J. Differential Equations, 249 (2010), 1746. doi: 10.1016/j.jde.2010.07.007. [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. doi: 10.4171/JEMS/83. [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. doi: 10.1142/S0129055X02001168. [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. doi: 10.1016/j.na.2003.05.001. [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. [7] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. [8] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations,, Adv. Nonlinear Stud., 2 (2002), 177. [9] M. J. Esteban and P.-L. Lions, A compactness lemma,, Nonlinear Analysis, 7 (1983), 381. [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. [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. [12] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, 65 (1986). [13] M. Willem, "Minimax Theorems,'', Progress in Nonlinear Differential Equations and their Applications, 24 (1996).

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. [2] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity,, J. Differential Equations, 249 (2010), 1746. doi: 10.1016/j.jde.2010.07.007. [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. doi: 10.4171/JEMS/83. [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. doi: 10.1142/S0129055X02001168. [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. doi: 10.1016/j.na.2003.05.001. [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. [7] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. [8] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations,, Adv. Nonlinear Stud., 2 (2002), 177. [9] M. J. Esteban and P.-L. Lions, A compactness lemma,, Nonlinear Analysis, 7 (1983), 381. [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. [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. [12] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, 65 (1986). [13] M. Willem, "Minimax Theorems,'', Progress in Nonlinear Differential Equations and their Applications, 24 (1996).
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