Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MULTIPLE SOLUTIONS OF A PERTURBED YAMABE-TYPE EQUATION ON GRAPH

  • Liu, Yang (Department of Mathematics Renmin University of China)
  • Received : 2021.12.10
  • Accepted : 2022.07.11
  • Published : 2022.09.01
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Abstract

Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ1(Ω) be constants, where λ1(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀0 > 0 such that for all 𝜀 ∈ (0, 𝜀0), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

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References

  1. Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in ℝN and its applications, Int. Math. Res. Not. IMRN 2010 (2010), no. 13, 2394-2426. https://doi.org/10.1093/imrn/rnp194
  2. S. Akduman and A. Pankov, Nonlinear Schrodinger equation with growing potential on infinite metric graphs, Nonlinear Anal. 184 (2019), 258-272. https://doi.org/10.1016/j.na.2019.02.020
  3. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  4. J. M. B. do O, N-Laplacian equations in ℝN with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3-4, 301-315. https://doi.org/10.1155/S1085337597000419
  5. J. M. do O, E. Medeiros, and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in ℝN, J. Differential Equations 246 (2009), no. 4, 1363-1386. https://doi.org/10.1016/j.jde.2008.11.020
  6. H. Ge and W. Jiang, Kazdan-Warner equation on infinite graphs, J. Korean Math. Soc. 55 (2018), no. 5, 1091-1101. https://doi.org/10.4134/JKMS.j170561
  7. H. Ge and W. Jiang, The 1-Yamabe equation on graphs, Commun. Contemp. Math. 21 (2019), no. 8, 1850040, 10 pp. https://doi.org/10.1142/S0219199718500402
  8. 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
  9. 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
  10. 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
  11. X. Han, M. Shao, and L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs, J. Differential Equations 268 (2020), no. 7, 3936-3961. https://doi.org/10.1016/j.jde.2019.10.007
  12. P. Horn, Y. Lin, S. Liu, and S. Yau, Volume doubling, Poincare inequality and Gaussian heat kernel estimate for non-negatively curved graphs, J. Reine Angew. Math. 757 (2019), 89-130. https://doi.org/10.1515/crelle-2017-0038
  13. S. Hou, Multiple solutions of a nonlinear biharmonic equation on graphs, preprint, 2021.
  14. 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
  15. M. Keller and M. Schwarz, The Kazdan-Warner equation on canonically compactifiable graphs, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 70, 18 pp. https://doi.org/10.1007/s00526-018-1329-7
  16. 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
  17. C. Liu and L. Zuo, Positive solutions of Yamabe-type equations with function coefficients on graphs, J. Math. Anal. Appl. 473 (2019), no. 2, 1343-1357. https://doi.org/10.1016/j.jmaa.2019.01.025
  18. S. Man, On a class of nonlinear Schrodinger equations on finite graphs, Bull. Aust. Math. Soc. 101 (2020), no. 3, 477-487. https://doi.org/10.1017/s0004972720000143
  19. C. Tian, Q. Zhang, and L. Zhang, Global stability in a networked SIR epidemic model, Appl. Math. Lett. 107 (2020), 106444, 6 pp. https://doi.org/10.1016/j.aml.2020.106444
  20. Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 262 (2012), no. 4, 1679-1704. https://doi.org/10.1016/j.jfa.2011.11.018
  21. X. Zhang and A. Lin, Positive solutions of p-th Yamabe type equations on infinite graphs, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1421-1427. https://doi.org/10.1090/proc/14362
  22. N. Zhang and L. Zhao, Convergence of ground state solutions for nonlinear Schrodinger equations on graphs, Sci. China Math. 61 (2018), no. 8, 1481-1494. https://doi.org/10.1007/s11425-017-9254-7