# American Institute of Mathematical Sciences

February  2019, 39(2): 819-839. doi: 10.3934/dcds.2019034

## On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces

 Department of Mathematics, Kyung Hee University, Seoul, 130-701, Korea

Received  December 2017 Revised  August 2018 Published  November 2018

In this paper, we study the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. The solution structure depends on the parameter $\varepsilon$ appearing in the equation. We find an upper bound $\varepsilon _c$ of $\varepsilon$ for the existence of solutions. By using the topological degree theory, we prove that there exist at least two solutions for $0<\varepsilon <\varepsilon _c$. We also study the asymptotic behavior of solutions as $\varepsilon \to 0$.

Citation: Jongmin Han, Juhee Sohn. On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 819-839. doi: 10.3934/dcds.2019034
##### References:
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##### References:
 [1] Y. Almog, Arbitrary n-vortex self-duality solutions to the Ginzburg-Lanbdu equations satisfying normal state conditions at infinity, Asymptotic Anal., 17 (1998), 267-278. Google Scholar [2] T. Aubin, Nonlinear Analysis on Manifolds: Monge-Ampére Equations, Springer-Velarg, Berline, 1982. doi: 10.1007/978-1-4612-5734-9. Google Scholar [3] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. P.D.E., 1 (1993), 123-148. doi: 10.1007/BF01191614. Google Scholar [4] E. Bogomol'nyi, The stability of classical solutions, Sov. J. Nucl. Phys., 24 (1976), 449-454. Google Scholar [5] L. A. Caffarelli and Y. Yang, Vortex condensation in Chern-Simons-Higgs model: An existence theorem, Comm. Math. Phys., 168 (1995), 321-336. Google Scholar [6] D. Chae, Global existence of solutions to the coupled einstein and maxwell-higgs system in the spherical symmetry, Ann. Henri Poincaré, 4 (2003), 35-62. doi: 10.1007/s00023-003-0121-0. Google Scholar [7] D. Chae, On the multi-string solutions of the self-dual static Einstein-Maxwell-Higgs system, Calc. Var. PDE, 20 (2004), 47-63. doi: 10.1007/s00526-003-0227-8. Google Scholar [8] X. Chen, S. Hastings, J. B. McLeod and Y. Yang, A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. Royal Soc. A, 446 (1994), 453-478. doi: 10.1098/rspa.1994.0115. Google Scholar [9] K. Choe, Multivortex solutions in the Chern imons gauged $O(3)$ sigma model on a doubly periodic domain, J. Math. Anal. Appl., 421 (2015), 591-624. doi: 10.1016/j.jmaa.2014.07.022. Google Scholar [10] K. Choe and N. Kim, Blow-up solutions of the self-dual Chern imons iggs vortex equation, Ann. Inst. Henri. Poincar - Anal. Nonlin., 25 (2008), 313-338. doi: 10.1016/j.anihpc.2006.11.012. Google Scholar [11] A. Comtet and G. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719-733. doi: 10.1016/0550-3213(88)90370-7. Google Scholar [12] G. Folland, Fourier Analysis and its Applications, Brooks/Cole, 1992. Google Scholar [13] J. Han, Asymptotics for the vortex condensate solutions in Chern-Simons-Higgs theory, Asymptotic Anal., 28 (2001), 31-48. Google Scholar [14] J. Han, Asymptotic limit for condensate solutions in the Abelian Chern-Simons Higgs model Ⅱ, Proc. Amer. Math. Soc., 131 (2003), 3827-3832. doi: 10.1090/S0002-9939-03-07020-5. Google Scholar [15] J. Han and C.-S. Lin, Multiplicity for self-dual condensate solutions in the Maxwell-Chern-Simons O(3) sigma model, Comm. PDE, 39 (2014), 1424-1450. doi: 10.1080/03605302.2014.908909. Google Scholar [16] J. Han and J. Sohn, Classification of string solutions for the self-dual Einstein-Maxwell-Higgs model, Preprint.Google Scholar [17] J. Han and J. Sohn, Existence of topological multi-string solutions in Abelian gauge field theories, J. Math. Phys., 58 (2017), 111511, 17 pp. doi: 10.1063/1.4997983. Google Scholar [18] M. Hindmarsh and T. Kibble, Cosmic strings, Rep. Prog. Phys., 58 (1995), 477-562. Google Scholar [19] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkhäuser, Boston, 1980. Google Scholar [20] B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativity & Gravitation, 20 (1988), 451-456. doi: 10.1007/BF00758120. Google Scholar [21] J. Spruck and Y. Yang, Regular stationary solutions of the cylindrically symmetric Einsteinmatter-gauge equations, J. Math. Anal. Appl., 195 (1995), 160-190. doi: 10.1006/jmaa.1995.1349. Google Scholar [22] G. Tarantello, Multiple condensate solutions for the Chern-Simons Higgs theory, J. Math. Phys., 37 (1996), 3769-3796. doi: 10.1063/1.531601. Google Scholar [23] G. Tarantello, Selfdual Gauge Field Vortices, Birkhuser, 2008. doi: 10.1007/978-0-8176-4608-0. Google Scholar [24] C. Taubes, Arbitrary $N$-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys., 72 (1980), 277-292. Google Scholar [25] S. Wang and Y. Yang, Abrikosov's vortices in the critical coupling, SIAM J. Math. Anal., 23 (1992), 1125-1140. doi: 10.1137/0523063. Google Scholar [26] Y. Yang, An equivalence theorem for string solutions of the Einstein matter-gauge equations, Lett. Math. Phys., 90 (1992), 79-90. doi: 10.1007/BF00398804. Google Scholar [27] Y. Yang, Prescribing topological defects for the coupled Einstein and abelian Higgs equations, Comm. Math. Phys., 170 (1995), 541-582. Google Scholar [28] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9. Google Scholar
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