American Institute of Mathematical Sciences

May  2012, 32(5): 1747-1761. doi: 10.3934/dcds.2012.32.1747

Solitary waves in critical Abelian gauge theories

 1 Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received  November 2010 Revised  June 2011 Published  January 2012

We prove existence of standing waves solutions for electrostatic Klein-Gordon-Maxwell systems in arbitrary dimensional compact Riemannian manifolds with boundary for zero Dirichlet boundary conditions. We prove that phase compensation holds true when the dimension $n = 3$ or $4$. In these dimensions, existence of a solution is obtained when the mass of the particle field, balanced by the phase, is small in a geometrically quantified sense. In particular, existence holds true for sufficiently large phases. When $n \ge 5$, existence of a solution is obtained when the mass of the particle field is sufficiently small.
Citation: Emmanuel Hebey. Solitary waves in critical Abelian gauge theories. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1747-1761. doi: 10.3934/dcds.2012.32.1747
References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [2] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar [3] 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. Google Scholar [4] _____, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar [5] T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball,, J. Differential Equations, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009. Google Scholar [6] _____, Clustered solutions around harmonic centers to a coupled elliptic system,, Ann. Inst. H. Poincaré Anal. Non Lin\'eaire, 24 (2007), 605. Google Scholar [7] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269. Google Scholar [8] P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 (2002). Google Scholar [9] P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. Google Scholar [10] _____, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009). Google Scholar [11] A. Azzollini, P. D'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779. Google Scholar [12] A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449. doi: 10.1017/S0308210509001814. Google Scholar [13] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. Google Scholar [14] _____, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, Topol. Methods Nonlinear Anal., 35 (2010), 33. Google Scholar [15] P. Bechouche, N. J. Mauser and S. Selberg, Nonrelativistic limit of Klein-Gordon-Maxwell to Schrödinger-Poisson,, Amer. J. Math., 126 (2004), 31. doi: 10.1353/ajm.2004.0001. Google Scholar [16] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409. doi: 10.1142/S0129055X02001168. Google Scholar [17] _____, Solitary waves in the nonlinear wave equation and in gauge theories,, J. Fixed Point Theory Appl., 1 (2007), 61. doi: 10.1007/s11784-006-0008-z. Google Scholar [18] _____, Solitary waves in abelian gauge theories,, Adv. Nonlinear Stud., 8 (2008), 327. Google Scholar [19] _____, 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. Google Scholar [20] _____, Hylomorphic vortices in abelian gauge theories,, preprint, (2009). Google Scholar [21] _____, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations,, Commun. Math. Phys., 295 (2010), 639. doi: 10.1007/s00220-010-0985-z. Google Scholar [22] H. Brézis and L. Nirenberg, Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [23] 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. Google Scholar [24] Y. Choquet-Bruhat, Solution globale des équations de Maxwell-Dirac-Klein-Gordon,, Rend. Circ. Mat. Palermo (2), 31 (1982), 267. doi: 10.1007/BF02844359. Google Scholar [25] E. Deumens, The Klein-Gordon-Maxwell nonlinear system of equations,, Solitons and Coherent Structures (Santa Barbara, 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0. Google Scholar [26] O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 19 (2002), 125. Google Scholar [27] O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831. doi: 10.1142/S0219199710004007. Google Scholar [28] O. Druet, E. Hebey and J. Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian,, J. Funct. Anal., 258 (2010), 999. doi: 10.1016/j.jfa.2009.07.004. Google Scholar [29] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness,, Comm. Math. Phys., 83 (1982), 171. doi: 10.1007/BF01976040. Google Scholar [30] V. Georgiev and N. Visciglia, Solitary waves for the Klein-Gordon-Maxwell system with external Coulomb potential,, J. Math. Pures Appl. (9), 84 (2005), 957. doi: 10.1016/j.matpur.2004.09.016. Google Scholar [31] E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in $4$-dimensional closed manifolds,, J. Reine Angew. Math., (). Google Scholar [32] E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds,, Duke Math. J., 79 (1995), 235. doi: 10.1215/S0012-7094-95-07906-X. Google Scholar [33] _____, Meilleures constantes dans le théorème d'inclusion de Sobolev,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 57. Google Scholar [34] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573. Google Scholar [35] M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm,, preprint, (2009). Google Scholar [36] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [37] S. Klainerman and D. Tataru, On the optimal regularity for Yang-Mills equations in $\mathbbR ^{4+1}$,, J. Amer. Math. Soc., 12 (1999), 93. doi: 10.1090/S0894-0347-99-00282-9. Google Scholar [38] E. Long, Existence and stability of solitary waves in non-linear Klein-Gordon-Maxwell equations,, Rev. Math. Phys., 18 (2006), 747. doi: 10.1142/S0129055X06002784. Google Scholar [39] E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law,, Rev. Math. Phys., 21 (2009), 459. doi: 10.1142/S0129055X09003669. Google Scholar [40] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations,, J. Amer. Math. Soc., 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [41] N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Comm. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0. Google Scholar [42] _____, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger,, Int. Math. Res. Not., 2003 (2003), 697. Google Scholar [43] 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. doi: 10.1098/rspa.2003.1267. Google Scholar [44] D. Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 27 (2010), 1055. Google Scholar [45] D. M. Petrescu, Time decay of solutions of coupled Maxwell-Klein-Gordon equations,, Commun. Math. Phys., 179 (1996), 11. doi: 10.1007/BF02103714. Google Scholar [46] I. Rodnianski and T. Tao, Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions,, Comm. Math. Phys., 251 (2004), 377. doi: 10.1007/s00220-004-1152-1. Google Scholar [47] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. doi: 10.1142/S0218202505003939. Google Scholar [48] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,, J. Differential Geom., 20 (1984), 479. Google Scholar [49] S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in $1+4$ dimensions,, Comm. Part. Diff. Eq., 27 (2002), 1183. Google Scholar [50] S. Selberg and A. Tesfahun, On the Maxwell-Klein-Gordon equations in Lorenz gauge,, Proceedings of the International Congress of Mathematical Physics, (2009). Google Scholar [51] W. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [52] T. Tao, Global behaviour of nonlinear dispersive and wave equations, in "Current Developments in Mathematics,", 2006, (2008), 255. Google Scholar

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References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [2] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar [3] 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. Google Scholar [4] _____, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar [5] T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball,, J. Differential Equations, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009. Google Scholar [6] _____, Clustered solutions around harmonic centers to a coupled elliptic system,, Ann. Inst. H. Poincaré Anal. Non Lin\'eaire, 24 (2007), 605. Google Scholar [7] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269. Google Scholar [8] P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 (2002). Google Scholar [9] P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. Google Scholar [10] _____, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009). Google Scholar [11] A. Azzollini, P. D'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779. Google Scholar [12] A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449. doi: 10.1017/S0308210509001814. Google Scholar [13] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. Google Scholar [14] _____, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, Topol. Methods Nonlinear Anal., 35 (2010), 33. Google Scholar [15] P. Bechouche, N. J. Mauser and S. Selberg, Nonrelativistic limit of Klein-Gordon-Maxwell to Schrödinger-Poisson,, Amer. J. Math., 126 (2004), 31. doi: 10.1353/ajm.2004.0001. Google Scholar [16] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409. doi: 10.1142/S0129055X02001168. Google Scholar [17] _____, Solitary waves in the nonlinear wave equation and in gauge theories,, J. Fixed Point Theory Appl., 1 (2007), 61. doi: 10.1007/s11784-006-0008-z. Google Scholar [18] _____, Solitary waves in abelian gauge theories,, Adv. Nonlinear Stud., 8 (2008), 327. Google Scholar [19] _____, 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. Google Scholar [20] _____, Hylomorphic vortices in abelian gauge theories,, preprint, (2009). Google Scholar [21] _____, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations,, Commun. Math. Phys., 295 (2010), 639. doi: 10.1007/s00220-010-0985-z. Google Scholar [22] H. Brézis and L. Nirenberg, Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [23] 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. Google Scholar [24] Y. Choquet-Bruhat, Solution globale des équations de Maxwell-Dirac-Klein-Gordon,, Rend. Circ. Mat. Palermo (2), 31 (1982), 267. doi: 10.1007/BF02844359. Google Scholar [25] E. Deumens, The Klein-Gordon-Maxwell nonlinear system of equations,, Solitons and Coherent Structures (Santa Barbara, 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0. Google Scholar [26] O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 19 (2002), 125. Google Scholar [27] O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831. doi: 10.1142/S0219199710004007. Google Scholar [28] O. Druet, E. Hebey and J. Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian,, J. Funct. Anal., 258 (2010), 999. doi: 10.1016/j.jfa.2009.07.004. Google Scholar [29] D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness,, Comm. Math. Phys., 83 (1982), 171. doi: 10.1007/BF01976040. Google Scholar [30] V. Georgiev and N. Visciglia, Solitary waves for the Klein-Gordon-Maxwell system with external Coulomb potential,, J. Math. Pures Appl. (9), 84 (2005), 957. doi: 10.1016/j.matpur.2004.09.016. Google Scholar [31] E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in $4$-dimensional closed manifolds,, J. Reine Angew. Math., (). Google Scholar [32] E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds,, Duke Math. J., 79 (1995), 235. doi: 10.1215/S0012-7094-95-07906-X. Google Scholar [33] _____, Meilleures constantes dans le théorème d'inclusion de Sobolev,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 57. Google Scholar [34] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573. Google Scholar [35] M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm,, preprint, (2009). Google Scholar [36] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4. Google Scholar [37] S. Klainerman and D. Tataru, On the optimal regularity for Yang-Mills equations in $\mathbbR ^{4+1}$,, J. Amer. Math. Soc., 12 (1999), 93. doi: 10.1090/S0894-0347-99-00282-9. Google Scholar [38] E. Long, Existence and stability of solitary waves in non-linear Klein-Gordon-Maxwell equations,, Rev. Math. Phys., 18 (2006), 747. doi: 10.1142/S0129055X06002784. Google Scholar [39] E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law,, Rev. Math. Phys., 21 (2009), 459. doi: 10.1142/S0129055X09003669. Google Scholar [40] M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations,, J. Amer. Math. Soc., 17 (2004), 297. doi: 10.1090/S0894-0347-03-00445-4. Google Scholar [41] N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Comm. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0. Google Scholar [42] _____, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger,, Int. Math. Res. Not., 2003 (2003), 697. Google Scholar [43] 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. doi: 10.1098/rspa.2003.1267. Google Scholar [44] D. Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials,, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 27 (2010), 1055. Google Scholar [45] D. M. Petrescu, Time decay of solutions of coupled Maxwell-Klein-Gordon equations,, Commun. Math. Phys., 179 (1996), 11. doi: 10.1007/BF02103714. Google Scholar [46] I. Rodnianski and T. Tao, Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions,, Comm. Math. Phys., 251 (2004), 377. doi: 10.1007/s00220-004-1152-1. Google Scholar [47] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. doi: 10.1142/S0218202505003939. Google Scholar [48] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,, J. Differential Geom., 20 (1984), 479. Google Scholar [49] S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in $1+4$ dimensions,, Comm. Part. Diff. Eq., 27 (2002), 1183. Google Scholar [50] S. Selberg and A. Tesfahun, On the Maxwell-Klein-Gordon equations in Lorenz gauge,, Proceedings of the International Congress of Mathematical Physics, (2009). Google Scholar [51] W. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [52] T. Tao, Global behaviour of nonlinear dispersive and wave equations, in "Current Developments in Mathematics,", 2006, (2008), 255. Google Scholar
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