Existence of Solutions to a Generalized Self-dual Chern-Simons Equation on Graphs

Abstract

Let G = (V,E) be a connected finite graph and Δ be the usual graph Laplacian. In this paper, the authors consider a generalized self-dual Chern-Simons equation on the graph G

(0.1)$\Delta u = - \lambda {{\rm{e}}^{F( u )}}{[ {{{\rm{e}}^{F( u )}} - 1} ]^2} + 4\pi \sum\limits_{j = 1}^M {{\delta _{{p_j}}}},$

where

$F(u) =\begin{cases}{\tilde F\left( u \right)}, & u \le 0,\\0, & u > 0,\end{cases}$

$\tilde F(u)$ satisfies $u=1+\tilde F(u)-\rm{{e}}^{\tilde F(u)}$, λ > 0, M is any fixed positive integer, $\delta_{p_{j}}$ is the Dirac delta mass at the vertex p_j, and p₁, p₂, …, p_M are arbitrarily chosen distinct vertices on the graph. They first prove that there is a critical value λ_c such that if λ ≥ λ_c, then the generalized self-dual Chern-Simons equation has a solution u_λ. Applying the existence result, they develop a new method to construct a solution of (0.1) which is monotonic with respect to λ when λ ≥ λ_c. Then they establish that there exist at least two solutions of the equation for λ > λ_c via the variational method. Furthermore, they give a fine estimate of the monotone solution, which can be applied to other related problems.

Keywords

Chern-Simons equation / Finite graph / Existence / Variational method / 35A01 / 35A15 / 35R02

References

[1] Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J. Functional Analysis, 1973, 14: 349-381.

[2] Bezryadina A, Eugenieva E, Chen Z. Self-trapping and flipping of double-charged vortices in optically induced photonic lattices. Optim. Lett., 2006, 31: 2456-2458.

[3] Chae D, Imanuvilov O Y. The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. Comm. Math. Phys., 2000, 215(1): 119-142.

[4] Chae D, Imanuvilov O Y. Non-topological solutions in the generalized self-dual Chern-Simons-Higgs theory. Calc. Var. Partial Differential Equations, 2003, 16(1): 47-61.

[5] Chan H, Fu C C, Lin C S. Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation. Comm. Math. Phys., 2002, 231(2): 189-221.

[6] Chen X, Hastings S, McLeod J B, Yang Y. A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. Roy. Soc. London Ser. A, 1994, 446(1928): 453-478.

[7] Ge H. Kazdan-Warner equation on graph in the negative case. J. Math. Anal. Appl., 2017, 453(2): 1022-1027.

[8] Ge H, Jiang W. Kazdan-Warner equation on infinite graphs. J. Korean Math. Soc., 2018, 55(5): 1091-1101

[9] Grigor’yan, A., Lin, Y. and Yang, Y., Kazdan-Warner equation on graph, Calc. Var. Partial Differential Equations, 55(4), 2016, Paper No. 92, 13 pp.

[10] Grigor’yan A, Lin Y, Yang Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci. China Math., 2017, 60(7): 1311-1324.

[11] Han X. Existence of doubly periodic vortices in a generalized Chern-Simons model. Nonlinear Anal. Real World Appl., 2014, 16: 90-102.

[12] Hong J, Kim Y, Pac P Y. Multivortex solutions of the abelian Chern-Simons-Higgs theory. Phys. Rev. Lett, 1990, 64(19): 2230-2233.

[13] Hou, S. and Sun, J., Existence of solutions to Chern-Simons-Higgs equations on graphs, Calc. Var. Partial Differential Equations, 61(4), 2022, Paper No. 139, 13 pp.

[14] Huang A, Lin Y, Yau S T. Existence of solutions to mean field equations on graphs. Comm. Math. Phys., 2020, 377(1): 613-621.

[15] Jaffe A, Taubes C. Vortices and Monopoles, Structure of static gauge theories, 1980. Boston, MA, Birkhäuser. 2

[16] Kazdan J L, Warner F W. Curvature functions for compact 2-manifolds. Ann. of Math, 1974, 99(2): 14-47.

[17] Keller, M. and Schwarz, M., The Kazdan-Warner equation on canonically compactifiable graphs, Calc. Var. Partial Differential Equations, 57(2), 2018, Paper No. 70, 18 pp.

[18] Nielsen H B, Olesen P. Vortex-line models for dual strings. Nucl. Phys. B, 1973, 61: 45-61.

[19] Sokoloff J B. Charged vortex excitations in quantum Hall systems. Phys. Rev. B, 1985, 31: 1924-1928.

[20] Spruck J, Yang Y. The existence of nontopological solitons in the self-dual Chern-Simons theory. Comm. Math. Phys., 1992, 149(2): 361-376.

[21] Spruck J, Yang Y. Topological solutions in the self-dual Chern-Simons theory: Existence and approximation. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 1995, 12(1): 75-97.

[22] Tarantello G. Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys., 1996, 37(8): 3769-3796.

[23] Tchrakian D H, Yang Y. The existence of generalised self-dual Chern-Simons vortices. Lett. Math. Phys., 1996, 36(4): 403-413.

[24] Wang R. The existence of Chern-Simons vortices. Comm. Math. Phys., 1991, 137(3): 587-597.

[25] Yang Y. Chern-Simons solitons and a nonlinear elliptic equation. Helv. Phys. Acta, 1998, 71(5): 573-585

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