Abstract
We prove strong convergence of the Bopp–Podolsky–Schrödinger–Proca system to the Schrödinger–Poisson–Proca system in the electro-magneto-static case as the Bopp–Podolsky parameter goes to zero.
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Aubin, T.: Espaces de Sobolev sur les variétés riemanniennes. Bull. Sci. Math. 100, 149–173 (1976)
Azzollini, A., d’Avenia, P., Pomponio, A.: On the Schrödinger–Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaré, Anal. Non Linéaire 27, 779–791 (2010)
Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)
Benci, V., Fortunato, D.: Spinning \(Q\)-balls for the Klein–Gordon–Maxwell equations. Commun. Math. Phys. 295, 639–668 (2010)
Bopp, F.: Eine Lineare Theorie des Elektrons. Ann. Phys. 430, 345–384 (1940)
Cuzinatto, R.R., De Morais, E.M., Medeiros, L.G., Naldoni de Souza, C., Pimentel, B.M.: De Broglie–Proca and Bopp–Podolsky massive photon gases in cosmology. EPL 118, 19001 (2017)
d’Avenia, P., Medreski, J., Pomponio, P.: Vortex ground states for Klein–Gordon–Maxwell–Proca type systems. J. Math. Phys. (to appear)
d’Avenia, P., Siciliano, G.: Nonlinear Schrödinger equation in the Bopp–Podolsky electrodynamics: solutions in the electrostatic case. Preprint
Dodziuk, J.: Sobolev spaces of differential forms and de Rham–Hodge isomorphism. J. Differ. Geom. 16, 63–73 (1981)
Druet, O.: From one bubble to several bubbles: the low-dimensional case. J. Differ. Geom. 63, 399–473 (2003)
Druet, O.: Compactness for Yamabe metrics in low dimensions. Int. Math. Res. Not. 23, 1143–1191 (2004)
Figueiredo, G.M., Siciliano, G.: Existence and asymptotic behaviour of solutions for a quasi-linear Schrödinger–Poisson system under a critical nonlinearity. Preprint (2017)
Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Part. Differ. Equ. 6, 883–901 (1981)
Gilbarg, G., Trüdinger, N.S.: Elliptic partial differential equations of second order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin (1983)
Hebey, E.: Compactness and Stability for Nonlinear Elliptic Equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich (2014)
Hebey, E.: Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting. Discrete Contin. Dyn. Syst. Ser. A 39, 6683–6712 (2019)
Hebey, E., Thizy, P.D.: Klein–Gordon–Maxwell–Proca type systems in the electro-magneto-static case. J. Part. Differ. Equ. 31, 119–58 (2018)
Hebey, E., Wei, J.: Schrödinger–Poisson systems in the 3-sphere. Calc. Var. Partial Differ. Equ. 47, 25–54 (2013)
Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Commun. Contemp. Math. 1, 1–50 (1999)
Marques, F.C.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. J. Differ. Geom. 71, 315–346 (2005)
Podolsky, B.: A generalized electrodynamics. Phys. Rev. 62, 68–71 (1942)
Schoen, R.M.: Lecture notes from courses at Stanford, written by D.Pollack. Preprint (1988)
Schoen, R.M., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)
Schrödinger, E.: The Earth’s and the Sun’s permanent magnetic fields in the unitary field theory. Proc. R. Ir. Acad. A 49, 135–148 (1943)
Thizy, P.D.: Non-resonant states for Schrödinger–Poisson critical systems in high dimensions. Arch. Math. 104, 485–490 (2015)
Thizy, P.D.: Schrödinger–Poisson systems in 4-dimensional closed manifolds. Discrete Contin. Dyn. Syst. Ser. A 36, 2257–2284 (2016)
Thizy, P.D.: Blow-up for Schrödinger–Poisson critical systems in dimensions 4 and 5. Calc. Var. Partial Differ. Equ. 55, 20 (2016)
Thizy, P.D.: Phase-stability for Schrödinger–Poisson critical systems in closed 5-manifolds. Int. Math. Res. Not. IMRN 20, 6245–6292 (2016)
Thizy, P.D.: Unstable phases for the critical Schrödinger–Poisson system in dimension 4. Differ. Integral Equa. 30, 825–832 (2017)
Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981)
Zayats, A.E.: Self-interaction in the Bopp–Podolsky electrodynamics: can the observable mass of a charged particle depend on its acceleration? Ann. Phys. 342, 11–20 (2014)
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The author would like to thank Michal Wrochna and Vladimir Georgescu for very interesting discussions, comments and very helpful suggestions. The author would also like to thank the referee for his/her constructive suggestions which definitely helped to improve the readability of the manuscript.
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Communicated by A. Malchiodi.
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Hebey, E. Strong convergence of the Bopp–Podolsky–Schrödinger–Proca system to the Schrödinger–Poisson–Proca system in the electro-magneto-static case. Calc. Var. 59, 198 (2020). https://doi.org/10.1007/s00526-020-01864-9
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DOI: https://doi.org/10.1007/s00526-020-01864-9