# 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 & Continuous Dynamical Systems - A, 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. Google Scholar [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.  Google Scholar [3] D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [15] Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar [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.  Google Scholar [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.  Google Scholar [18] L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar [19] D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar [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.  Google Scholar [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. Google Scholar

show all references

##### 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. Google Scholar [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.  Google Scholar [3] D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [15] Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar [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.  Google Scholar [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.  Google Scholar [18] L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar [19] D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar [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.  Google Scholar [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. Google Scholar
 [1] Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $BV$ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405 [2] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400 [3] Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172 [4] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [5] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [6] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [7] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [8] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [9] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 [10] Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002 [11] Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278 [12] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 [13] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [14] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [15] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 [16] Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 [17] Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407 [18] Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2021001 [19] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [20] Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041

2019 Impact Factor: 1.338