# American Institute of Mathematical Sciences

October  2021, 20(10): 3373-3393. doi: 10.3934/cpaa.2021109

## Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model

 1 Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2 California State University, Los Angeles, 5151, USA

* Corresponding author

Received  December 2020 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: This work is supported by NNSF of China, No: 12071189 and 12001252, by the Jiangxi Provincial Natural Science Foundation, 20202BAB201005 and No: 20202ACBL201001, by the Science and Technology Research Project of Jiangxi Provincial Department of Education, No: 200307 and 200325

Our purpose in this paper is to classify the non-topological solutions of equations
 $-\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E)$
where
 $\{\delta_{p_i}\}_{i = 1}^k$
(resp.
 $\{\delta_{q_j}\}_{j = 1}^l$
) are Dirac masses concentrated at the points
 $\{p_i\}_{i = 1}^k$
, (resp.
 $\{q_j\}_{j = 1}^l$
),
 $n_i$
and
 $m_j$
are positive integers. Denote
 $N = \sum^k_{i = 1}n_i$
and
 $M = \sum^l_{j = 1}m_j$
satisfying that
 $N-M>1$
.
Problem
 $(E)$
arises from gauged sigma models and we first construct an extremal non-topological solution
 $u$
of
 $(E)$
with asymptotic behavior
 $u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty$
and with total magnetic flux
 $4\pi (N-M-1)$
. And then we do the classification for non-topological solutions of
 $(E)$
with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.
Citation: Huyuan Chen, Hichem Hajaiej. Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3373-3393. doi: 10.3934/cpaa.2021109
##### References:
 [1] R. Beeker, Electromagnetic Fields and Interactions, Dover, New York, 1982. [2] A. Belavin and A. Polyakov, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247. [3] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $\Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253. doi: 10.1080/03605309108820797. [4] M. Cantor, Elliptic operators and the decomposition of tensor fields, Bull. Amr. Math. Soc., 5 (1981), 235-262. doi: 10.1090/S0273-0979-1981-14934-X. [5] M. Chae, Existence of multi-string solutions of the gauged harmonic map model, Lett. Math. Phys., 59 (2002), 173-188. doi: 10.1023/A: 1014912714390. [6] H. Chan, C. Fu and C. Lin, Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221. doi: 10.1007/s00220-002-0691-6. [7] H. Chen, H. Hajaiej, L. Veron, Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model, arXiv: 2002.02685. doi: 10.1016/j. na. 2021.112257. [8] H. Chen and F. Zhou, Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet, Nonlinear Anal., 196 (2020), 111788. doi: 10.1016/j. na. 2020.111788. [9] K. Cheng and C. Lin, On the Conformal Gaussian Curvature Equation in $\mathbb{R}^2$, J. Differ. Equ., 146 (1998), 226-250. doi: 10.1006/jdeq. 1998.3424. [10] K. Cheng and C. Lin, Conformal metrics with prescribed nonpositive Gaussian on $\mathbb{R}^2$, Calc. Var. Partial Differ. Equ., 11 (2000), 203-231. doi: 10.1007/s005260000037. [11] K. Cheng and W. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbb{R}^2$, Duke Math. J., 62 (1991), 721-737. doi: 10.1215/S0012-7094-91-06231-9. [12] K. Cheng and W. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbb{R}^2$ II, Math. Ann, 290 (1991), 671-680. doi: 10.1007/BF01459266. [13] J. Chern and Z. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, J. Funct. Anal., 265 (2013), 1240-1263. doi: 10.1016/j. jfa. 2013.05.041. [14] N. Choi and J. Han, Classification of solutions of elliptic equations arising from a gravitational $O(3)$ gauge field model, J. Differ. Equ., 264 (2018), 4944-4988. doi: 10.1016/j. jde. 2017.12.030. [15] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin/New York, 1977. [16] J. Han and H. Huh, Existence of topological solutions in the Maxwell gauged $O(3)$ sigma models, J. Math. Anal. Appl., 386 (2012), 61-74. doi: 10.1016/j. jmaa. 2011.07.046. [17] W. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv., 34 (1960), 75-84. doi: 10.1007/BF02565929. [18] A. Jaffe and C. Taubes, Vortices and Monoples, Birkhäuser, Boston, 1980. [19] J. Jost and G. Wang, Analytic aspects of the Toda system: I. A Moser-Trudinger inequality, Commun. Pure Appl. Math., 54 (2001), 1289-1319. doi: 10.1002/cpa. 10004. [20] J. B. Keller, On solutions of $\Delta u = f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa. 3160100402. [21] F. Lin and Y. Yang, Gauged harmonic maps, Born-Infeld electromagnetism, and magnetic vortices, Commun. Pure Appl. Math., 56 (2003), 1631-1665. doi: 10.1002/cpa. 10106. [22] C. Lin, J. Wei and D. Ye, Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207. doi: 10.1007/s00222-012-0378-3. [23] R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Commun. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa. 3160320604. [24] R. Osserman, On the inequality $\Delta u = f(u)$, Pac. J. Math., 7 (1957), 1641-1647. [25] A. Poliakovsky and G. Tarantello, On non-topological solutions for planar Liouville Systems of Toda-type, Commun. Math. Phys., 347 (2016), 223-270. doi: 10.1007/s00220-016-2662-3. [26] R. Rajaraman, Solitons and Instantons, Amsterdam: North Holland, 1982. [27] B. Schroers, Bogomol'nyi solitons in a gauged $O(3)$ sigma model, Phys. Lett. B., 356 (1995), 291-296. doi: 10.1016/0370-2693(95)00833-7. [28] K. Song, Improved existence results of solutions to the gravitational Maxwell gauged $O(3)$ sigma model, Proc. Amer. Math. Soc., 144 (2016), 3499-3505. doi: 10.1090/proc/12967. [29] Y. Wang and H. Chen, On anisotropic singularities for semi-linear elliptic equations in $\mathbb{R}^2$, J. Math. Anal. Appl., 451 (2017), 931-953. doi: 10.1016/j. jmaa. 2017.02.045. [30] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-6548-9. [31] Y. Yang, A necessary and sufficient conditions for the existence of multisolitons in a self-dual gauged sigma model, Commun. Math. Phys., 181 (1996), 485-506. [32] Y. Yang, The Existence of Solitons in Gauged Sigma Models with Broken Symmetry: Some Remarks, Lett. Math. Phys., 40 (1997), 177-189. doi: 10.1023/A: 1007363726173. [33] L. Véron, Elliptic Equations Involving Measures, Stationary Partial Differential Equations, North-Holland, Amsterdam, 2004. doi: 10.1016/S1874-5733(04)80010-X.

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##### References:
 [1] R. Beeker, Electromagnetic Fields and Interactions, Dover, New York, 1982. [2] A. Belavin and A. Polyakov, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247. [3] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $\Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253. doi: 10.1080/03605309108820797. [4] M. Cantor, Elliptic operators and the decomposition of tensor fields, Bull. Amr. Math. Soc., 5 (1981), 235-262. doi: 10.1090/S0273-0979-1981-14934-X. [5] M. Chae, Existence of multi-string solutions of the gauged harmonic map model, Lett. Math. Phys., 59 (2002), 173-188. doi: 10.1023/A: 1014912714390. [6] H. Chan, C. Fu and C. Lin, Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221. doi: 10.1007/s00220-002-0691-6. [7] H. Chen, H. Hajaiej, L. Veron, Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model, arXiv: 2002.02685. doi: 10.1016/j. na. 2021.112257. [8] H. Chen and F. Zhou, Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet, Nonlinear Anal., 196 (2020), 111788. doi: 10.1016/j. na. 2020.111788. [9] K. Cheng and C. Lin, On the Conformal Gaussian Curvature Equation in $\mathbb{R}^2$, J. Differ. Equ., 146 (1998), 226-250. doi: 10.1006/jdeq. 1998.3424. [10] K. Cheng and C. Lin, Conformal metrics with prescribed nonpositive Gaussian on $\mathbb{R}^2$, Calc. Var. Partial Differ. Equ., 11 (2000), 203-231. doi: 10.1007/s005260000037. [11] K. Cheng and W. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbb{R}^2$, Duke Math. J., 62 (1991), 721-737. doi: 10.1215/S0012-7094-91-06231-9. [12] K. Cheng and W. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbb{R}^2$ II, Math. Ann, 290 (1991), 671-680. doi: 10.1007/BF01459266. [13] J. Chern and Z. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, J. Funct. Anal., 265 (2013), 1240-1263. doi: 10.1016/j. jfa. 2013.05.041. [14] N. Choi and J. Han, Classification of solutions of elliptic equations arising from a gravitational $O(3)$ gauge field model, J. Differ. Equ., 264 (2018), 4944-4988. doi: 10.1016/j. jde. 2017.12.030. [15] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin/New York, 1977. [16] J. Han and H. Huh, Existence of topological solutions in the Maxwell gauged $O(3)$ sigma models, J. Math. Anal. Appl., 386 (2012), 61-74. doi: 10.1016/j. jmaa. 2011.07.046. [17] W. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv., 34 (1960), 75-84. doi: 10.1007/BF02565929. [18] A. Jaffe and C. Taubes, Vortices and Monoples, Birkhäuser, Boston, 1980. [19] J. Jost and G. Wang, Analytic aspects of the Toda system: I. A Moser-Trudinger inequality, Commun. Pure Appl. Math., 54 (2001), 1289-1319. doi: 10.1002/cpa. 10004. [20] J. B. Keller, On solutions of $\Delta u = f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa. 3160100402. [21] F. Lin and Y. Yang, Gauged harmonic maps, Born-Infeld electromagnetism, and magnetic vortices, Commun. Pure Appl. Math., 56 (2003), 1631-1665. doi: 10.1002/cpa. 10106. [22] C. Lin, J. Wei and D. Ye, Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207. doi: 10.1007/s00222-012-0378-3. [23] R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Commun. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa. 3160320604. [24] R. Osserman, On the inequality $\Delta u = f(u)$, Pac. J. Math., 7 (1957), 1641-1647. [25] A. Poliakovsky and G. Tarantello, On non-topological solutions for planar Liouville Systems of Toda-type, Commun. Math. Phys., 347 (2016), 223-270. doi: 10.1007/s00220-016-2662-3. [26] R. Rajaraman, Solitons and Instantons, Amsterdam: North Holland, 1982. [27] B. Schroers, Bogomol'nyi solitons in a gauged $O(3)$ sigma model, Phys. Lett. B., 356 (1995), 291-296. doi: 10.1016/0370-2693(95)00833-7. [28] K. Song, Improved existence results of solutions to the gravitational Maxwell gauged $O(3)$ sigma model, Proc. Amer. Math. Soc., 144 (2016), 3499-3505. doi: 10.1090/proc/12967. [29] Y. Wang and H. Chen, On anisotropic singularities for semi-linear elliptic equations in $\mathbb{R}^2$, J. Math. Anal. Appl., 451 (2017), 931-953. doi: 10.1016/j. jmaa. 2017.02.045. [30] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-6548-9. [31] Y. Yang, A necessary and sufficient conditions for the existence of multisolitons in a self-dual gauged sigma model, Commun. Math. Phys., 181 (1996), 485-506. [32] Y. Yang, The Existence of Solitons in Gauged Sigma Models with Broken Symmetry: Some Remarks, Lett. Math. Phys., 40 (1997), 177-189. doi: 10.1023/A: 1007363726173. [33] L. Véron, Elliptic Equations Involving Measures, Stationary Partial Differential Equations, North-Holland, Amsterdam, 2004. doi: 10.1016/S1874-5733(04)80010-X.
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