Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vertex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.
Abstract
Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vertex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.
참고문헌 (23)
- A. A. Abrikosov, "On the magnetic properties of superconductors of the second group", Sov. Phys. JETP, 5, pp.1174~1182 (1957)
- F. Bauer, "Li-Yau inequality on graphs", J. Differential Geom, 99(3), pp.359~405 (2015)
- D. Bazeia, "Generalized self-dual Chern-Simons vortices", Phys. Rev. D, 81, pp.125014~ (2010)
- E. Bendito, "Solving boundary value problems on networks using equilibrium measures", J. Funct. Anal, 171(1), pp.155~176 (2000)[DOI]
- L. A. Caffarelli, "Vortex condensation in the Chern-Simons Higgs model : an existence theorem", Comm. Math. Phys, 168(2), pp.321~336 (1995)
- D. Chae, "Non-topological solutions in the generalized self-dual Chern-Simons-Higgs theory", Calc. Var. Partial Differential Equations, 16(1), pp.47~61 (2003)[DOI]
- H. Ge, "A note on Liouville type equations on graphs", Proc. Amer. Math. Soc, 146(11), pp.4837~4842 (2018)[DOI]
- H. Ge, "Kazdan-Warner equation on infinite graphs", J. Korean Math. Soc, 55(5), pp.1091~1101 (2018)[DOI]
- H. Ge, "The 1-Yamabe equation on graphs", Commun. Contemp. Math, 21(8), pp.10~ (2019)[DOI]
- A. Grigor’yan, "Kazdan-Warner equation on graph", Calc. Var. Partial Differential Equations, 55(4), pp.13~ (2016)[DOI]
- X. Han, "The existence of multi-vortices for a generalized self-dual Chern-Simons model", Nonlinearity, 26(3), pp.805~835 (2013)[DOI]
- P. Horn, "Volume doubling, Poincar´e inequality and Gaussian heat kernel estimate for non-negatively curved graphs", J. Reine Angew. Math, 757, pp.89~130 (2019)[DOI]
- S. Hou, "Existence of solutions to Chern-Simons-Higgs equations on graphs", Calc. Var. Partial Differential Equations, 61(4), pp.13~ (2022)[DOI]
- A. Huang, "Existence of solutions to mean field equations on graphs", Comm. Math. Phys, 377(1), pp.613~621 (2020)[DOI]
- Y. Lin, "The existence and nonexistence of global solutions for a semilinear heat equation on graphs", Calc. Var. Partial Differential Equations, 56(4), pp.22~ (2017)[DOI]
- Y. Liu, "Brouwer degree for mean field equation on graph", Bull. Korean Math. Soc, 59(5), pp.1305~1315 (2022)[DOI]
- H. B. Nielsen, "Vortex line models for dual strings", Nuclear Phys. B, 61, pp.45~61 (1973)
- L. Sun, "Brouwer degree for Kazdan-Warner equations on a connected finite graph", Adv. Math, 404(part B), pp.29~ (2022)[DOI]
- G. Tarantello, "Multiple condensate solutions for the Chern-Simons-Higgs theory", J. Math. Phys, 37(8), pp.3769~3796 (1996)[DOI]
- D. H. Tchrakian, "The existence of generalised self-dual Chern-Simons vortices", Lett. Math. Phys, 36(4), pp.403~413 (1996)[DOI]
- Y. Wu, "Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs", Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 115(3), pp.16~ (2021)[DOI]
- Y. Yang, "Chern-Simons solitons and a nonlinear elliptic equation", Helv. Phys. Acta, 71(5), pp.573~585 (1998)