## Magneto-static vortices in two-dimensional Abelian gauge theories.(English)Zbl 1181.35227

Summary: 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 $${\mathbf A}$$ is achieved with the help of a penalization method.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q60 PDEs in connection with optics and electromagnetic theory 78A25 Electromagnetic theory (general)

### Keywords:

Klein-Gordon-Maxwell equations; two-dimensional vortex
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### References:

 [1] Abrikosov A.A.: On the magnetic properties of superconductors of the second group. Soviet Physics. JETP 5, 1174–1182 (1957) [2] Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) · Zbl 0273.49063 [3] Azzollini A., Benci V., D’Aprile T., Fortunato D.: Existence of static solutions of the semilinear Maxwell equations. Ric. Mat. 55, 283–297 (2006) · Zbl 1150.35078 [4] 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) · Zbl 0973.35161 [5] Benci V., Fortunato D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002) · Zbl 1037.35075 [6] Benci V., Fortunato D.: Towards a unified field theory for Classical Electrodynamics. Arch. Rat. Mech. Anal. 173, 379–414 (2004) · Zbl 1065.78004 [7] Benci V., Fortunato D.: Solitary waves in the nolinear wave equation and in gauge theories. J. Fixed Point Theory Appl. 1, 61–86 (2007) · Zbl 1122.35121 [8] V.Benci, D.Fortunato, Three dimensional vortices in Abelian Gauge Theories. ArXiv:0711.3351v1 [math.AP]. · Zbl 1173.81013 [9] 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) · Zbl 1064.35182 [10] Felsager B.: Geometry, Particles and Fields. Odense University Press, Odense (1981) · Zbl 0489.58001 [11] Fetter A.L., Walecka J.D.: Quantum Theory of Many-Particle Systems. Dover, New York (2003) · Zbl 1191.70001 [12] Nielsen H., Olesen P.: Vortex-line models for dual strings. Nuclear Phys. B 61, 45–61 (1973) [13] Palais R.S.: The principle of symmetric criticality. Comm. Math. Phys. 79, 19–30 (1979) · Zbl 0417.58007 [14] Rajaraman R.: Solitons and Instantons. North-Holland, Amsterdam (1989) [15] V. Rubakov, Classical Theory of Gauge Fields. Princeton University Press, 2002. · Zbl 1036.81002 [16] Yang Y.: Solitons in Field Theory and Nonlinear Analysis. Springer, New York, Berlin (2000)
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