Abstract
We study the existence of vortices of the Klein-Gordon-Maxwell equations in the two dimensional case. In particular we find sufficient conditions for the existence of vortices in the magneto-static case, i.e. when the electric potential \({\phi = 0}\). This result, due to the lack of suitable embedding theorems for the vector potential A is achieved with the help of a penalization method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrikosov A.A.: On the magnetic properties of superconductors of the second group. Soviet Physics. JETP 5, 1174–1182 (1957)
Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Azzollini A., Benci V., D’Aprile T., Fortunato D.: Existence of static solutions of the semilinear Maxwell equations. Ric. Mat. 55, 283–297 (2006)
Benci V., d’Avenia P., Fortunato D., Pisani L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)
Benci V., Fortunato D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)
Benci V., Fortunato D.: Towards a unified field theory for Classical Electrodynamics. Arch. Rat. Mech. Anal. 173, 379–414 (2004)
Benci V., Fortunato D.: Solitary waves in the nolinear wave equation and in gauge theories. J. Fixed Point Theory Appl. 1, 61–86 (2007)
V.Benci, D.Fortunato, Three dimensional vortices in Abelian Gauge Theories. ArXiv:0711.3351v1 [math.AP].
D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schroedinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 134, 893–906 (2004)
Felsager B.: Geometry, Particles and Fields. Odense University Press, Odense (1981)
Fetter A.L., Walecka J.D.: Quantum Theory of Many-Particle Systems. Dover, New York (2003)
Nielsen H., Olesen P.: Vortex-line models for dual strings. Nuclear Phys. B 61, 45–61 (1973)
Palais R.S.: The principle of symmetric criticality. Comm. Math. Phys. 79, 19–30 (1979)
Rajaraman R.: Solitons and Instantons. North-Holland, Amsterdam (1989)
V. Rubakov, Classical Theory of Gauge Fields. Princeton University Press, 2002.
Yang Y.: Solitons in Field Theory and Nonlinear Analysis. Springer, New York, Berlin (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are supported by MIUR - PRIN2005 “Metodi variazionali e topologici nello studio di fenomeni non lineari”.
Rights and permissions
About this article
Cite this article
Bellazzini, J., Bonanno, C. & Siciliano, G. Magneto-Static Vortices in Two Dimensional Abelian Gauge Theories. Mediterr. J. Math. 6, 347–366 (2009). https://doi.org/10.1007/s00009-009-0013-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-009-0013-8