Skip to main content
SpringerLink
Log in

Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we investigate a class of p-Laplacian systems on a locally finite graph \(G=(V,E)\). By exploiting the method of Nehari manifold and some new analytical techniques, under suitable assumptions on the potentials and nonlinear terms, we prove that the p-Laplacian system admits a ground state solution \((u_{\lambda },v_{\lambda })\) when the parameter \(\lambda \) is sufficiently large. Furthermore, we consider the concentration behavior of these solutions as \(\lambda \rightarrow \infty \), and show that these solutions converge to a solution of the corresponding limit problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Bose, N., Liang, P.: Neural Network Fundamentals with Graphs, Algorithms and Applications. McGraw-Hill, New York (1996)

    MATH  Google Scholar 

  2. Lezoray, O., Grady, L.: Image Processing and Analysis with Graphs: Theory and Practice. CRC Press, Boca Raton (2017)

    Book  MATH  Google Scholar 

  3. Grigor’yan, A., Lin, Y., Yang, Y.: Kazdan–Warner equation on graph. Calc. Var. Partial Differ. Equ. 55(4), 1–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grigor’yan, A., Lin, Y., Yang, Y.: Yamabe type equations on graphs. J. Differ. Equ. 261, 4924–4943 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17, 1047–1060 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Curtis, E., Morrow, J.: Determining the resistors in a network. SIAM J. Appl. Math. 50(3), 918–930 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, N., Zhao, L.: Convergence of ground state solutions for nonlinear Schrödinger equations on graphs. Sci. China Math. 61(8), 1481–1494 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Han, X., Shao, M.: \(p\)-Laplacian equations on locally finite graphs. Acta Math. Sin. (Engl. Ser.) 37(11), 1645–1678 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, X., Shao, M., Zhao, L.: Existence and convergence of solutions for nonlinear biharmonic equations on graphs. J. Differ. Equ. 268(7), 3936–3961 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Furtado, M.F.: Multiple minimal nodal solutions for a quasilinear Schrödinger equation with symmetric potential. J. Math. Anal. Appl. 304, 170–188 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gheraibia, B.: Solutions for quasilinear elliptic systems with vanishing potentials. Topol. Methods Nonlinear Anal. 54(1), 153–175 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Lü, D., Liu, Q.: Multiplicity of solutions for a class of quasilinear Schrödinger systems in \({\mathbb{R} }^{N}\). Comput. Math. Appl. 66(12), 2532–2544 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Morais Filho, D.C., Souto, M.A.S.: Systems of \(p\)-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Commun. Partial Differ. Equ. 24, 1537–1553 (1999)

    Article  MATH  Google Scholar 

  15. Huang, D., Li, Y.: Multiplicity of solutions for a noncooperative \(p\)-Laplacian elliptic system in \({\mathbb{R} }^{N}\). J. Differ. Equ. 215, 206–223 (2005)

    Article  MATH  Google Scholar 

  16. Bartsch, T., Wang, Z.Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366–384 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nakamura, T., Yamasaki, M.: Generalized extremal length of an infinite network. Hiroshima Math. J. 6, 95–111 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iannizzotto, A., Tersian, S.A.: Multiple homoclinic solutions for the discrete \(p\)-Laplacian via critical point theory. J. Math. Anal. Appl. 403, 173–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mugnolo, D.: Parabolic theory of the discrete \(p\)-Laplace operator. Nonlinear Anal. 87, 33–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xu, J., Zhao, L.: Existence and convergence of solutions for nonlinear elliptic systems on graphs. Commun. Math. Stat. (2022). https://doi.org/10.48550/arXiv.2111.11078

  21. Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.T.: Li–Yau inequality on graphs. J. Differ. Geom. 99(3), 359–405 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ge, H., Jiang, W.: Yamabe equations on infinite graphs. J. Math. Anal. Appl. 460, 885–890 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ge, H., Hua, B., Jiang, W.: A note on Liouville equations on graphs. Proc. Am. Math. Soc. 146, 4837–4842 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, X., Keller, M., Masamune, J., Wojciechowski, R.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265, 1556–1578 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, Y., Wu, Y.: The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc. Var. Partial Differ. Equ. 56(4), 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  26. Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257, 4133–4164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 493–516 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Duan, S., Wu, X.: An existence result for a class of \(p\)-Laplacian elliptic systems involving homogeneous nonlinearities in \({\mathbb{R} }^{N}\). Nonlinear Anal. 74, 4723–4737 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, Q., Yang, Z.: Multiplicity of positive solutions for a \((p, q)\)-Laplacian system with concave and critical nonlinearities. J. Math. Anal. Appl. 423, 660–680 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhao, X., Tang, C.L.: Resonance problems for \((p, q)\)-Laplacian systems. Nonlinear Anal. 72, 1019–1030 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, X.: Existence and multiplicity of weak solutions for a nonlinear impulsive \((q, p)\)-Laplacian dynamical system. Adv. Differ. Equ. 1, 1–12 (2017)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    Mengqiu Shao

Authors
  1. Mengqiu Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mengqiu Shao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by the NSFC (12101355).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shao, M. Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs. J. Fixed Point Theory Appl. 25, 50 (2023). https://doi.org/10.1007/s11784-023-01055-x

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11784-023-01055-x

Keywords

Mathematics Subject Classification

Search

Navigation