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Existence of Three Solutions to a Class of Nonlinear Equations on Graphs

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Abstract

Let u be a function on locally finite connect graph G = (V, E) and Ω be a bounded subset of V. We consider the nonlinear Dirichlet boundary condition problem

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = f(u)}&{\text{in}}&\Omega \\ {u = 0,\;\;\;\;\;\;}&{\text{on}}&{\partial \Omega .} \end{array}} \right.$$

Let f: ℝ → ℝ be a function satisfying certain assumptions. Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions, of which one is trivial and the others are strictly positive.

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References

  1. Adimurthi, Yang, Y.: An interpolation of Hardy inequality and Trudinger—Moser inequality in ℝN and its applications. Int. Math. Res. Not. IMRN, 13), 2394–2426 (2010)

    MATH  Google Scholar 

  2. Chang, K.-C.: Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005

    MATH  Google Scholar 

  3. Chung, F.: Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997

    MATH  Google Scholar 

  4. Chen, C.-C., Lin, C.-S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math., 56), 1667–1727 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grigor’yan, A., Lin, Y., Yang, Y.: Yamabe type equations on graphs. J. Differential Equations, 261), 4924–4943 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grigor’yan, A., Lin, Y., Yang, Y.: Kazdan–Warner equation on graph. Calc. Var. Partial Differ. Equ., 55, Paper No. 92, 13 pp. (2016)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Ge, H., Jiang, W.: The 1-Yamabe equation on graph. Commun. Contemp. Math., 21(8), 1850040, 10 pp. (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, X., Shao, M., Zhao, L.: Existence and convergence of solutions for nonlinear biharmonic equations on graphs. J. Differential Equations, 268), 3936–3961 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, A., Lin, Y., Yau, S.-T.: Existence of solutions to mean field equations on graphs. Commun. Math. Phys., 377), 613–621 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hou, S.: Multiple solutions of a nonlinear biharmonic equation on graphs, arXiv: 2205.07798 (2022)

  13. Keller, M., Schwarz, M.: The Kazdan–Warner equation on canonically compactifiable graphs. Calc. Var. Partial Differ. Equ., 57, Paper No. 70, 18 pp. (2018)

  14. Lin, Y., Yang, Y.: A heat flow for the mean field equation on a finite graph. Calc. Var. Partial Differ. Equ., 60(6), Paper No. 206, 15 pp. (2021)

  15. Lin, Y., Yang, Y.: Calculus of variations on locally finite graphs. Rev. Mat. Complut., 35(3), 791–813 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, S., Yang, Y.: Multiple solutions of Kazdan–Warner equation on graphs in the negative case. Calc. Var. Partial Differ. Equ., 59(5), Paper No. 164, 15 pp. (2020)

  17. Liu, C., Zuo, L.: Positive solutions of Yamabe-type equations with function coefficients on graphs. J. Math. Anal. Appl., 473), 1343–1357 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marcos do Ó, J., Medeiros, E., Severo, U.: On a quasilinear nonhomogeneous elliptic equation with critical growth in ℝN. J. Differential Equations, 246), 1363–1386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, L., Wang, L.: Brouwer degree for Kazdan–Warner equations on a connected finite graph. Adv. Math., 404, Paper No. 108422, 29 pp. (2022)

  20. Yang, Y.: Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. J. Funct. Anal., 262), 1679–1704 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, N., Zhao, L.: Convergence of ground state solutions for nonlinear Schrödinger equations on graphs. Sci. China Math., 61), 1481–1494 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, X., Lin, A.: Positive solutions of p-th Yamabe type equations on infinite graphs. Proc. Amer. Math. Soc., 147), 1421–1427 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu, X.: Mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs, J. Partial Differ. Equ., 35(3), 199–207 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks the referees for their time and helpful comments. Furthermore, the author is very grateful to Professor Yang Yunyan for his suggestions on the condition description of Theorem 1.2, and for providing the author with the three-solution theorem.

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  1. School of Mathematics, Renmin University of China, Beijing, 100872, P.R. China

    Yang Liu

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  1. Yang Liu
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Correspondence to Yang Liu.

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Liu, Y. Existence of Three Solutions to a Class of Nonlinear Equations on Graphs. Acta. Math. Sin.-English Ser. 39, 1129–1137 (2023). https://doi.org/10.1007/s10114-023-2142-6

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  • DOI: https://doi.org/10.1007/s10114-023-2142-6

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