Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, p. 1055-1071

We study the existence of radially symmetric solitary waves for a system of a nonlinear Klein–Gordon equation coupled with Maxwell's equation in presence of a positive mass. The nonlinear potential appearing in the system is assumed to be positive and with more than quadratical growth at infinity.

DOI : https://doi.org/10.1016/j.anihpc.2010.02.001
Classification:  35J50,  81T13
Keywords: Klein–Gordon–Maxwell system, Positive superquadratic potential, Lagrange multiplier, Nontrivial solutions
@article{AIHPC_2010__27_4_1055_0,
     author = {Mugnai, Dimitri},
     title = {Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {4},
     year = {2010},
     pages = {1055-1071},
     doi = {10.1016/j.anihpc.2010.02.001},
     zbl = {1194.35378},
     mrnumber = {2659157},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_4_1055_0}
}
Mugnai, Dimitri. Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, pp. 1055-1071. doi : 10.1016/j.anihpc.2010.02.001. http://www.numdam.org/item/AIHPC_2010__27_4_1055_0/

[1] V. Benci, D. Fortunato, Existence of hylomorphic solitary waves in Klein–Gordon and in Klein–Gordon–Maxwell equations, Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Rend Lincei (9) Mat. Appl. 20 (2009), 243-279 | MR 2540181 | Zbl 1194.35343

[2] V. Benci, D. Fortunato, Solitary waves in Abelian gauge theories, Adv. Nonlinear Stud. 8 no. 2 (2008), 327-352 | MR 2402825 | Zbl 1157.58005

[3] V. Benci, D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations, Rev. Math. Phys. 14 (2002), 409-420 | MR 1901222 | Zbl 1037.35075

[4] V. Benci, D. Fortunato, Towards a unified field theory for classical electrodynamics, Arch. Ration. Mech. Anal. 173 (2004), 379-414 | MR 2091510 | Zbl 1065.78004

[5] V. Benci, D. Fortunato, Three-dimensional vortices in abelian gauge theories, Nonlinear Anal. 70 (2009), 4402-4421 | MR 2514771 | Zbl 1173.81013

[6] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I Existence of a ground state, Arch. Ration. Mech. Anal. 82 no. 4 (1983), 313-345 | MR 695535 | Zbl 0533.35029

[7] T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 no. 4 (1982), 549-561 | MR 677997 | Zbl 0513.35007

[8] T. D'Aprile, D. Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud. 4 no. 3 (2004), 307-322 | MR 2079817 | Zbl 1142.35406

[9] T. D'Aprile, D. Mugnai, Solitary Waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 1-14 | MR 2099569 | Zbl 1064.35182

[10] P. D'Avenia, L. Pisani, Nonlinear Klein–Gordon equations coupled with Born–Infeld type equations, Electron. J. Differential Equations 26 (2002), 1-13 | MR 1884995 | Zbl 0993.35083

[11] K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York (1982) | MR 696935 | Zbl 0496.35001

[12] M.J. Esteban, V. Georgiev, E. Sere, Stationary waves of the Maxwell–Dirac and the Klein–Gordon–Dirac equations, Calc. Var. Partial Differential Equations 4 (1996), 265-281 | MR 1386737 | Zbl 0869.35105

[13] B. Felsager, Geometry, Particles and Fields, Odense University Press (1981) | MR 664923 | Zbl 0489.58001

[14] S. Klainerman, M. Machedon, On the Maxwell–Klein–Gordon equation with finite energy, Duke Math. J. 74 no. 1 (1994), 19-44 | MR 1271462 | Zbl 0818.35123

[15] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 no. 2 (1984), 109-145 | Numdam | Zbl 0541.49009

[16] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 no. 4 (1984), 223-283 | Numdam | MR 778974 | Zbl 0704.49004

[17] E. Long, Existence and stability of solitary waves in non-linear Klein–Gordon–Maxwell equations, Rev. Math. Phys. 18 (2006), 747-779 | MR 2267114 | Zbl 1169.78003

[18] M. Machedon, J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations, J. Amer. Math. Soc. 17 no. 2 (2004), 297-359 | MR 2051613 | Zbl 1048.35115

[19] D. Mugnai, Coupled Klein–Gordon and Born–Infeld type equations: looking for solitons, Proc. R. Soc. Lond. Ser. A 460 (2004), 1519-1528 | MR 2066416 | Zbl 1078.35100

[20] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, Oxford, New York, Tokyo (1988) | MR 719693 | Zbl 0694.35194

[21] I. Rodnianski, T. Tao, Global regularity for the Maxwell–Klein–Gordon equation with small critical Sobolev norm in high dimensions, Comm. Math. Phys. 251 no. 2 (2004), 377-426 | MR 2100060 | Zbl 1106.35073

[22] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 no. 2 (1977), 149-162 | Zbl 0356.35028

[23] C.H. Taubes, On the Yang–Mills–Higgs equations, Bull. Amer. Math. Soc. (N.S.) 10 no. 2 (1984), 295-297 | MR 733700 | Zbl 0551.35028

[24] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. vol. 24, Birkhäuser Boston, Inc., Boston, MA (1996) | MR 1400007 | Zbl 0856.49001