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Multiple results on nontrivial solutions of discrete Kirchhoff type problems

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Abstract

In this paper, we apply the variational technique together with the local linking theory or the fountain theorem to study a class of discrete Kirchhoff type problems with Dirichlet boundary conditions. Examples and numerical simulations are also provided to illustrate applications of our results.

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References

  1. Long, Y.H., Wang, L.: Global dynamics of a delayed two-patch discrete SIR disease model. Commun. Nonlinear Sci. Numer. Simul. 83, 105117 (2020)

    Article  MATH  Google Scholar 

  2. Shi, Y.T., Yu, J.S.: Wolbachia infection enhancing and decaying domains in mosquito population based on discrete models. J. Biol. Dyn. 14(1), 679–695 (2020)

    Article  Google Scholar 

  3. Lin, G.H., Zhou, Z., Yu, J.S.: Ground state solutions of discrete asymptotically linear Schrödinge equations with bounded and non-periodic potentials. J. Dyn. Diff. Equat. 32(2), 527–555 (2020)

    Article  MATH  Google Scholar 

  4. Zhou, Z., Ling, J.X.: Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \(\phi _{c}\)-Laplacian. Appl. Math. Lett. 91, 28–34 (2019)

    Article  Google Scholar 

  5. Long, Y.H., Chen, J.L.: Existence of multiple solutions to second-order discrete Neumann boundary value problems. Appl. Math. Lett. 83, 7–14 (2018)

    Article  MATH  Google Scholar 

  6. Long, Y.H.: Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients. J. Differ. Equ. Appl. 26(7), 966–986 (2020)

    Article  MATH  Google Scholar 

  7. Long, Y.H.: Existence of two homoclinic solutions for a nonperiodic difference equation with a perturbation. AIMS Math. 6(5), 4786–4802 (2021)

    Article  MATH  Google Scholar 

  8. Marchuk, G.J.: Methods of Numerical Mathematics, 2nd edn. Springer-Verlag, New York (1982)

    Book  MATH  Google Scholar 

  9. Cheng, S.S.: Partial Difference Equations. Taylor and Francis, London (2003)

    Book  MATH  Google Scholar 

  10. Du, S.J., Zhou, Z.: On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator. Adv. Nonlinear Anal. 11, 198–211 (2022)

    Article  MATH  Google Scholar 

  11. Imbesi, M., Bisci, G.M.: Discrete elliptic Dirichlet problems and nonlinear algebraic systems. Mediterr. J. Math. 13, 263–278 (2016)

    Article  MATH  Google Scholar 

  12. Long, Y.H., Zhang, H.: Three nontrivial solutions for second order partial difference equation via Morse theory. J. Funct. Spaces (2022) to appear

  13. Long, Y.H., Deng, X.Q.: Existence and multiplicity solutions for discrete Kirchhoff type problems. Appl. Math. Lett. 126, 107817 (2022)

    Article  MATH  Google Scholar 

  14. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

  15. Perera, K., Zhang, Z.T.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)

    Article  MATH  Google Scholar 

  16. Mao, A., Zhang, Z.T.: Sign-changing and multiple solutions of Kirchhoff type problems without P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)

    Article  MATH  Google Scholar 

  17. Sun, J.J., Tang, C.L.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212–1222 (2011)

    Article  MATH  Google Scholar 

  18. He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({ R^3}\). J. Differ. Equ. 252, 1813–1834 (2012)

    Article  MATH  Google Scholar 

  19. Wu, K., Zhou, F., Gu, G.Z.: Some remarks on uniqueness of positive solutions to Kirchhoff type equations. Appl. Math. Lett. 124, 107642 (2022)

    Article  MATH  Google Scholar 

  20. Ji, J., Yang, B.: Eigenvalue comparisons for boundary value problems of the discrete elliptic equation. Commun. Appl. Anal. 12(2), 189–198 (2008)

    MATH  Google Scholar 

  21. Yang, J.P., Liu, J.S.: Nontrivial solutions for discrete Kirchhoff-type problems with resonance via critical groups. Adv. Differ. Equ. 2013, 308 (2013)

    Article  MATH  Google Scholar 

  22. Liu, J.Q.: The Morse index of a saddle point. Systems. Sci. Math. Sci. 2, 32–39 (1998)

    MATH  Google Scholar 

  23. Zhang, J.H., Li, S.J.: Multiple nontrivial solutions for some fourth order semilinear elliptic problems. Nonlinear Anal. 60, 221–230 (2005)

    Article  MATH  Google Scholar 

  24. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Yuhua Long

  2. Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou, 510006, China

    Yuhua Long

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  1. Yuhua Long
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Correspondence to Yuhua Long.

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This work is supported by the National Natural Science Foundation of China (Grant No. 11971126).

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Long, Y. Multiple results on nontrivial solutions of discrete Kirchhoff type problems. J. Appl. Math. Comput. 69, 1–17 (2023). https://doi.org/10.1007/s12190-022-01731-0

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  • DOI: https://doi.org/10.1007/s12190-022-01731-0

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