# American Institute of Mathematical Sciences

June  2020, 40(6): 3291-3304. doi: 10.3934/dcds.2020127

## On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory

 a. Department of Mathematics, National Central University, Chung-Li 32001, Taiwan b. Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan

* Corresponding author; Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 107-2115-M-008-005-MY3

1 Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 105-2115-M-008-012-MY3.
2 Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 106-2628-M-018-001-MY4.
Dedicated to Professor Wei-Ming Ni in honor of his 70th birthday

Received  April 2019 Published  February 2020

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [18]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

Citation: Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
##### References:
 [1] W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. [2] D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119–142. doi: 10.1007/s002200000302. [3] D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. [4] W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in ${\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439. doi: 10.1215/S0012-7094-93-07117-7. [5] K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in ${\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250. doi: 10.1006/jdeq.1998.3424. [6] J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263. [7] J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp. doi: 10.1063/1.4867618. [8] K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. [9] K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852. doi: 10.1016/j.anihpc.2011.06.003. [10] K. Choe, N. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in ${\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37. [11] K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in ${\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp. doi: 10.1007/s00526-017-1119-7. [12] K. Choe, N. Kim, Y. Lin and C.-S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in ${\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012. [13] A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733. doi: 10.1016/0550-3213(88)90370-7. [14] A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. [15] Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. [16] G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769. [17] B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456. doi: 10.1007/BF00758120. [18] L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. [19] D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. [20] Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33. doi: 10.1016/j.nuclphysb.2014.05.013. [21] Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609.

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##### References:
 [1] W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. [2] D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119–142. doi: 10.1007/s002200000302. [3] D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. [4] W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in ${\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439. doi: 10.1215/S0012-7094-93-07117-7. [5] K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in ${\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250. doi: 10.1006/jdeq.1998.3424. [6] J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263. [7] J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp. doi: 10.1063/1.4867618. [8] K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. [9] K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852. doi: 10.1016/j.anihpc.2011.06.003. [10] K. Choe, N. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in ${\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37. [11] K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in ${\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp. doi: 10.1007/s00526-017-1119-7. [12] K. Choe, N. Kim, Y. Lin and C.-S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in ${\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012. [13] A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733. doi: 10.1016/0550-3213(88)90370-7. [14] A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. [15] Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. [16] G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769. [17] B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456. doi: 10.1007/BF00758120. [18] L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. [19] D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. [20] Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33. doi: 10.1016/j.nuclphysb.2014.05.013. [21] Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609.
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