대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제59권5호
  • /
  • Pages.1305-1315
  • /
  • 2022
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

BROUWER DEGREE FOR MEAN FIELD EQUATION ON GRAPH

  • Liu, Yang (Department of Mathematics Renmin University of China)
  • 투고 : 2021.10.15
  • 심사 : 2022.04.07
  • 발행 : 2022.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let u be a function on a connected finite graph G = (V, E). We consider the mean field equation (1) $-{\Delta}u={\rho}\({\frac{he^u}{\int_Vhe^ud{\mu}}}-{\frac{1}{{\mid}V{\mid}}}\),$ where ∆ is 𝜇-Laplacian on the graph, 𝜌 ∈ ℝ\{0}, h : V → ℝ+ is a function satisfying minx∈V h(x) > 0. Following Sun and Wang [15], we use the method of Brouwer degree to prove the existence of solutions to the mean field equation (1). Firstly, we prove the compactness result and conclude that every solution to the equation (1) is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation (1), say $$d_{{\rho},h}=\{{-1,\;{\rho}>0, \atop 1,\;{\rho}<0.}$$ Consequently, the equation (1) has at least one solution due to the Brouwer degree d𝜌,h ≠ 0.

키워드

참고문헌

  1. H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of -∆u =V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223-1253. https://doi.org/10.1080/03605309108820797
  2. K.-C. Chang, Methods in nonlinear analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
  3. C.-C. Chen and C.-S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (2003), no. 12, 1667-1727. https://doi.org/10.1002/cpa.10107
  4. F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.
  5. W. Ding, J. Jost, J. Li, and G. Wang, Existence results for mean field equations, Ann. Inst. H. Poincare C Anal. Non Lineaire 16 (1999), no. 5, 653-666. https://doi.org/10.1016/S0294-1449(99)80031-6
  6. Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. Math. 10 (2008), no. 2, 205-220. https://doi.org/10.1142/S0219199708002776
  7. A. Grigor'yan, Y. Lin, and Y. Yang, Yamabe type equations on graphs, J. Differential Equations 261 (2016), no. 9, 4924-4943. https://doi.org/10.1016/j.jde.2016.07.011
  8. A. Grigor'yan, Y. Lin, and Y. Yang, Kazdan-Warner equation on graph, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 92, 13 pp. https://doi.org/10.1007/s00526-016-1042-3
  9. A. Grigor'yan, Y. Lin, and Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math. 60 (2017), no. 7, 1311-1324. https://doi.org/10.1007/s11425-016-0422-y
  10. A. Huang, Y. Lin, and S.-T. Yau, Existence of solutions to mean field equations on graphs, Comm. Math. Phys. 377 (2020), no. 1, 613-621. https://doi.org/10.1007/s00220-020-03708-1
  11. Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), no. 2, 421-444. https://doi.org/10.1007/s002200050536
  12. Y. Lin and Y. Yang, A heat flow for the mean field equation on a finite graph, Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 206, 15 pp. https://doi.org/10.1007/s00526-021-02086-3
  13. S. Liu and Y. Yang, Multiple solutions of Kazdan-Warner equation on graphs in the negative case, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 164, 15 pp. https://doi.org/10.1007/s00526-020-01840-3
  14. M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 109-121.
  15. L. Sun and L. Wang, Brouwer degree for Kazdan-Warner equations on a connected finite graph, Adv. Math. 404 (2022), Paper No. 108422, 29 pp. https://doi.org/10.1016/j.aim.2022.108422
  16. X. Zhu, Mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs, J. Part. Diff. Eq. 35 (2022), no. 3, 199-207. https://doi.org/10.4208/jpde.v35.n3.1