# 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 and 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. [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. [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. [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. [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. [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. [7] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. [8] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [9] M. J. Esteban and P.-L. Lions, A compactness lemma, Nonlinear Analysis, 7 (1983), 381-385. [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. [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. [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. [13] M. Willem, "Minimax Theorems,'' Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, 1996.

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. [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. [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. [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. [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. [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. [7] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. [8] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [9] M. J. Esteban and P.-L. Lions, A compactness lemma, Nonlinear Analysis, 7 (1983), 381-385. [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. [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. [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. [13] M. Willem, "Minimax Theorems,'' Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, 1996.
 [1] Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097 [2] Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135 [3] 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 [4] Sitong Chen, Xianhua Tang. Improved results for Klein-Gordon-Maxwell systems with general nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2333-2348. doi: 10.3934/dcds.2018096 [5] Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034 [6] Norihisa Ikoma. Multiplicity of radial and nonradial solutions to equations with fractional operators. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3501-3530. doi: 10.3934/cpaa.2020153 [7] Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 [8] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [9] Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 503-518. doi: 10.3934/dcdss.2020028 [10] Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669 [11] Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146 [12] Hartmut Pecher. Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2965-2989. doi: 10.3934/cpaa.2021091 [13] M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573 [14] Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure and Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465 [15] Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024 [16] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 [17] Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315 [18] Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389 [19] Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889 [20] P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029

2021 Impact Factor: 1.588