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November  2019, 39(11): 6683-6712. doi: 10.3934/dcds.2019291

Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting

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

Received  March 2019 Revised  June 2019 Published  August 2019

We investigate the system consisting of the the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting in the context of closed $ 3 $-dimensional manifolds. We prove existence of solutions up to the gauge, and compactness of the system both in the subcritical and in the critical case.

Citation: Emmanuel Hebey. Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6683-6712. doi: 10.3934/dcds.2019291
References:
[1]

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

[2]

T. Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sc. Math., 100 (1976), 149-173.   Google Scholar

[3]

V. Benci and D. Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Commun. Math. Phys., 295 (2010), 639-668.  doi: 10.1007/s00220-010-0985-z.  Google Scholar

[4]

F. Bopp, Eine lineare Theorie des Elektrons, Ann. Phys., 38 (1940), 345-384.  doi: 10.1002/andp.19404300504.  Google Scholar

[5]

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.  doi: 10.1090/S0894-0347-07-00575-9.  Google Scholar

[6]

S. Brendle and F. C. Marques, Blow-up phenomena for the Yamabe equation Ⅱ, J. Differential Geom., 81 (2009), 225-250.  doi: 10.4310/jdg/1231856261.  Google Scholar

[7]

___, Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics, 20 (2011), 29–47. Google Scholar

[8]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[9]

P. d'Avenia, J. Medreski and P. Pomponio, Vortex ground states for Klein-Gordon-Maxwell-Proca type systems, J. Math. Phys., 58 (2017), 041503, 19 pp. doi: 10.1063/1.4982038.  Google Scholar

[10]

P. d'Avenia and G. Siciliano, Nonlinear Schrödinger equation in thje Bopp-Podolsky electrodynamics: solutions in the electrostatic case, J. Differential Equations, 267 (2019), 1025-1065.  doi: 10.1016/j.jde.2019.02.001.  Google Scholar

[11]

J. Dodziuk, Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Diff. Geom., 16 (1981), 63-73.  doi: 10.4310/jdg/1214435988.  Google Scholar

[12]

O. Druet, From one bubble to several bubbles: The low-dimensional case, J. Differential Geom., 63 (2003), 399-473.  doi: 10.4310/jdg/1090426771.  Google Scholar

[13]

___, Compactness for Yamabe metrics in low dimensions, Internat. Math. Res. Notices, 23 (2004), 1143–1191. doi: 10.1155/S1073792804133278.  Google Scholar

[14]

O. Druet and E. Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Analysis and PDEs, 2 (2009), 305-359.  doi: 10.2140/apde.2009.2.305.  Google Scholar

[15]

___, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z., 263 (2009), 33–67. doi: 10.1007/s00209-008-0409-3.  Google Scholar

[16]

___., Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831–869. doi: 10.1142/S0219199710004007.  Google Scholar

[17]

O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, Princeton University Press, vol. 45, 2004. doi: 10.1007/BF01158557.  Google Scholar

[18]

O. DruetE. 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), 999-1059.  doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[19]

___, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds. Ⅱ, J. reine angew. Math., 713 (2016), 149–179. Google Scholar

[20]

P. EspositoA. Pistoia and J. Vétois, The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[21]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Part. Diff. Eq., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[22]

G. Gilbarg and N. S. Trüdinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin–New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[23]

A. S. Goldhaber and M. M. Nieto, Terrestrial and Extraterrestrial limits on the photon mass, Rev. Mod. Phys., 43 (1971), 277-296.   Google Scholar

[24]

___, Photon and Graviton mass limits, Rev. Mod. Phys., 82 (2010), 939–979. Google Scholar

[25]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2014. doi: 10.4171/134.  Google Scholar

[26]

E. Hebey and P. D. Thizy, Stationary Kirchhoff systems in closed $3$-dimensional manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2085-2114.  doi: 10.1007/s00526-015-0858-6.  Google Scholar

[27]

___, Klein-Gordon-Maxwell-Proca type systems in the electro-magneto-static case, J. Part. Diff. Eq., 31 (2018), 119–58. Google Scholar

[28]

E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279.  doi: 10.1215/S0012-7094-95-07906-X.  Google Scholar

[29]

___, Meilleures constantes dans le théorème d'inclusion de Sobolev, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 57–93. doi: 10.1016/S0294-1449(16)30097-X.  Google Scholar

[30]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the $3$-sphere, Calc. Var. Partial Dif- ferential Equations, 47 (2013), 25-54.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[31]

M. KhuriF. C. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.  doi: 10.4310/jdg/1228400630.  Google Scholar

[32]

Y. Y. Li and L. Zhang, A Harnack type inequality for the Yamabe equations in low dimensions, Calc. Var. PDE, 20 (2004), 133-151.  doi: 10.1007/s00526-003-0230-0.  Google Scholar

[33]

___, Compactness of solutions to the Yamabe problem Ⅱ, Calc. Var. PDE, 24 (2005), 185–237. Google Scholar

[34]

___, Compactness of solutions to the Yamabe problem Ⅲ, J. Funct. Anal., 245 (2007), 438–474. doi: 10.1016/j.jfa.2006.11.010.  Google Scholar

[35]

Y. Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[36]

J. LuoG. T. Gillies and L. C. Tu, The mass of the photon, Rep. Prog. Phys., 68 (2005), 77-130.   Google Scholar

[37]

F. C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.  doi: 10.4310/jdg/1143651772.  Google Scholar

[38]

B. Podolsky, A Generalized Electrodynamics, hys. Rev., 62 (1942), 68-71.   Google Scholar

[39]

H. Ruegg and M. Ruiz-Altaba, The Stueckelberg field, Int. J. Mod. Phys. A, 19 (2004), 3265-3347.  doi: 10.1142/S0217751X04019755.  Google Scholar

[40]

R. M. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.  doi: 10.4310/jdg/1214439291.  Google Scholar

[41]

___, Lecture Notes from Courses at Stanford, written by D.Pollack, preprint, 1988. Google Scholar

[42]

___, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math., Springer-Verlag, Berlin, 1365 (1989), 120–154. doi: 10.1007/BFb0089180.  Google Scholar

[43]

___, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A Symposium in Honor of Manfredo do Carmo, Proc. Int. Conf. (Rio de Janeiro, 1988)., Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 311–320.  Google Scholar

[44]

___, A report on some recent progress on nonlinear problems in geometry, Surveys in Differential Geometry (Cambridge, 1990), Suppl. J. Diff. Geom., Lehigh University, Pennsylvania, 1 (1991), 201–241. Google Scholar

[45]

R. M. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.  doi: 10.1007/BF01940959.  Google Scholar

[46]

P. D. Thizy, Non-resonant states for Schrödinger-Poisson critical systems in high dimensions, Arch. Math., 104 (2015), 485-490.  doi: 10.1007/s00013-015-0763-4.  Google Scholar

[47]

___, Schrödinger-Poisson systems in 4-dimensional closed manifolds, Discrete Contin. Dyn. Syst.-Series A, 36 (2016), 2257–2284. doi: 10.3934/dcds.2016.36.2257.  Google Scholar

[48]

___, Blow-up for Schrödinger-Poisson critical systems in dimensions $4$ and $5$, Calc. Var. Partial Differential Equations, 55 (2016), Art. 20, 21 pp. doi: 10.1007/s00526-016-0959-x.  Google Scholar

[49]

___, Phase-stability for Schrödinger-Poisson critical systems in closed 5-manifolds, Int. Math. Res. Not. IMRN, 20 (2016), 6245–6292. doi: 10.1093/imrn/rnv344.  Google Scholar

[50]

___, Unstable phases for the critical Schrödinger-Poisson system in dimension 4, Differential Integral Equations, 30 (2017), 825–832.  Google Scholar

[51]

N. S. Trüdinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.   Google Scholar

[52]

E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys., 80 (1981), 381-402.  doi: 10.1007/BF01208277.  Google Scholar

show all references

References:
[1]

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

[2]

T. Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sc. Math., 100 (1976), 149-173.   Google Scholar

[3]

V. Benci and D. Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Commun. Math. Phys., 295 (2010), 639-668.  doi: 10.1007/s00220-010-0985-z.  Google Scholar

[4]

F. Bopp, Eine lineare Theorie des Elektrons, Ann. Phys., 38 (1940), 345-384.  doi: 10.1002/andp.19404300504.  Google Scholar

[5]

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.  doi: 10.1090/S0894-0347-07-00575-9.  Google Scholar

[6]

S. Brendle and F. C. Marques, Blow-up phenomena for the Yamabe equation Ⅱ, J. Differential Geom., 81 (2009), 225-250.  doi: 10.4310/jdg/1231856261.  Google Scholar

[7]

___, Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics, 20 (2011), 29–47. Google Scholar

[8]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[9]

P. d'Avenia, J. Medreski and P. Pomponio, Vortex ground states for Klein-Gordon-Maxwell-Proca type systems, J. Math. Phys., 58 (2017), 041503, 19 pp. doi: 10.1063/1.4982038.  Google Scholar

[10]

P. d'Avenia and G. Siciliano, Nonlinear Schrödinger equation in thje Bopp-Podolsky electrodynamics: solutions in the electrostatic case, J. Differential Equations, 267 (2019), 1025-1065.  doi: 10.1016/j.jde.2019.02.001.  Google Scholar

[11]

J. Dodziuk, Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Diff. Geom., 16 (1981), 63-73.  doi: 10.4310/jdg/1214435988.  Google Scholar

[12]

O. Druet, From one bubble to several bubbles: The low-dimensional case, J. Differential Geom., 63 (2003), 399-473.  doi: 10.4310/jdg/1090426771.  Google Scholar

[13]

___, Compactness for Yamabe metrics in low dimensions, Internat. Math. Res. Notices, 23 (2004), 1143–1191. doi: 10.1155/S1073792804133278.  Google Scholar

[14]

O. Druet and E. Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Analysis and PDEs, 2 (2009), 305-359.  doi: 10.2140/apde.2009.2.305.  Google Scholar

[15]

___, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z., 263 (2009), 33–67. doi: 10.1007/s00209-008-0409-3.  Google Scholar

[16]

___., Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831–869. doi: 10.1142/S0219199710004007.  Google Scholar

[17]

O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, Princeton University Press, vol. 45, 2004. doi: 10.1007/BF01158557.  Google Scholar

[18]

O. DruetE. 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), 999-1059.  doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[19]

___, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds. Ⅱ, J. reine angew. Math., 713 (2016), 149–179. Google Scholar

[20]

P. EspositoA. Pistoia and J. Vétois, The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.  doi: 10.1007/s00208-013-0971-9.  Google Scholar

[21]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Part. Diff. Eq., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[22]

G. Gilbarg and N. S. Trüdinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin–New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[23]

A. S. Goldhaber and M. M. Nieto, Terrestrial and Extraterrestrial limits on the photon mass, Rev. Mod. Phys., 43 (1971), 277-296.   Google Scholar

[24]

___, Photon and Graviton mass limits, Rev. Mod. Phys., 82 (2010), 939–979. Google Scholar

[25]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2014. doi: 10.4171/134.  Google Scholar

[26]

E. Hebey and P. D. Thizy, Stationary Kirchhoff systems in closed $3$-dimensional manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2085-2114.  doi: 10.1007/s00526-015-0858-6.  Google Scholar

[27]

___, Klein-Gordon-Maxwell-Proca type systems in the electro-magneto-static case, J. Part. Diff. Eq., 31 (2018), 119–58. Google Scholar

[28]

E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279.  doi: 10.1215/S0012-7094-95-07906-X.  Google Scholar

[29]

___, Meilleures constantes dans le théorème d'inclusion de Sobolev, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 57–93. doi: 10.1016/S0294-1449(16)30097-X.  Google Scholar

[30]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the $3$-sphere, Calc. Var. Partial Dif- ferential Equations, 47 (2013), 25-54.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[31]

M. KhuriF. C. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.  doi: 10.4310/jdg/1228400630.  Google Scholar

[32]

Y. Y. Li and L. Zhang, A Harnack type inequality for the Yamabe equations in low dimensions, Calc. Var. PDE, 20 (2004), 133-151.  doi: 10.1007/s00526-003-0230-0.  Google Scholar

[33]

___, Compactness of solutions to the Yamabe problem Ⅱ, Calc. Var. PDE, 24 (2005), 185–237. Google Scholar

[34]

___, Compactness of solutions to the Yamabe problem Ⅲ, J. Funct. Anal., 245 (2007), 438–474. doi: 10.1016/j.jfa.2006.11.010.  Google Scholar

[35]

Y. Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[36]

J. LuoG. T. Gillies and L. C. Tu, The mass of the photon, Rep. Prog. Phys., 68 (2005), 77-130.   Google Scholar

[37]

F. C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.  doi: 10.4310/jdg/1143651772.  Google Scholar

[38]

B. Podolsky, A Generalized Electrodynamics, hys. Rev., 62 (1942), 68-71.   Google Scholar

[39]

H. Ruegg and M. Ruiz-Altaba, The Stueckelberg field, Int. J. Mod. Phys. A, 19 (2004), 3265-3347.  doi: 10.1142/S0217751X04019755.  Google Scholar

[40]

R. M. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.  doi: 10.4310/jdg/1214439291.  Google Scholar

[41]

___, Lecture Notes from Courses at Stanford, written by D.Pollack, preprint, 1988. Google Scholar

[42]

___, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math., Springer-Verlag, Berlin, 1365 (1989), 120–154. doi: 10.1007/BFb0089180.  Google Scholar

[43]

___, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A Symposium in Honor of Manfredo do Carmo, Proc. Int. Conf. (Rio de Janeiro, 1988)., Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 311–320.  Google Scholar

[44]

___, A report on some recent progress on nonlinear problems in geometry, Surveys in Differential Geometry (Cambridge, 1990), Suppl. J. Diff. Geom., Lehigh University, Pennsylvania, 1 (1991), 201–241. Google Scholar

[45]

R. M. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.  doi: 10.1007/BF01940959.  Google Scholar

[46]

P. D. Thizy, Non-resonant states for Schrödinger-Poisson critical systems in high dimensions, Arch. Math., 104 (2015), 485-490.  doi: 10.1007/s00013-015-0763-4.  Google Scholar

[47]

___, Schrödinger-Poisson systems in 4-dimensional closed manifolds, Discrete Contin. Dyn. Syst.-Series A, 36 (2016), 2257–2284. doi: 10.3934/dcds.2016.36.2257.  Google Scholar

[48]

___, Blow-up for Schrödinger-Poisson critical systems in dimensions $4$ and $5$, Calc. Var. Partial Differential Equations, 55 (2016), Art. 20, 21 pp. doi: 10.1007/s00526-016-0959-x.  Google Scholar

[49]

___, Phase-stability for Schrödinger-Poisson critical systems in closed 5-manifolds, Int. Math. Res. Not. IMRN, 20 (2016), 6245–6292. doi: 10.1093/imrn/rnv344.  Google Scholar

[50]

___, Unstable phases for the critical Schrödinger-Poisson system in dimension 4, Differential Integral Equations, 30 (2017), 825–832.  Google Scholar

[51]

N. S. Trüdinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.   Google Scholar

[52]

E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys., 80 (1981), 381-402.  doi: 10.1007/BF01208277.  Google Scholar

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