Skip to main content
SpringerLink
Log in

Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we are interested in the existence of infinitely many solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, we determine unbounded intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. Finally, the attained solutions are positive when the nonlinearity is supposed to be nonnegative thanks to a discrete maximum principle.

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

References

  1. Agarwal R.P.: Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York-Basel (2000)

  2. Agarwal R.P., Perera K., O’Regan D.: Multiple positive solutions of singular discrete p-Laplacian problems via variational methods. Adv. Differ. Equ. 2, 93–99 (2005)

    MathSciNet  Google Scholar 

  3. Bereanu C., Mawhin J.: Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions. Math. Bohem. 131, 145–160 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Candito P., Molica Bisci G.: Existence of solutions for a nonlinear algebraic system with a parameter. Appl. Math. Comput. 218, 11700–11707 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng S.S.: Partial Difference Equations. Taylor & Francis, London (2003)

    Book  MATH  Google Scholar 

  6. Chu J., Jiang D.: Eingenvalues and discrete boundary value problems for the one-dimensional p-Laplacian. J. Math. Anal. 305, 452–465 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Galewski M., Orpel A.: On the existence of solutions for discrete elliptic boundary value problems. Appl. Anal. 89, 1879–1891 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henderson J., Thompson H.B.: Existence of multiple solutions for second order discrete boundary value problems. Comput. Math. Appl. 43, 1239–1248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ji J., Yang B.: Eigenvalue comparisons for boundary value problems of the discrete elliptic equation. Commun. Appl. Anal. 12(2), 189–197 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Kelly W.G., Peterson A.C.: Difference equations, an introduction with applications. Academic Press, San Diego (1991)

    Google Scholar 

  11. Kristály, A., Rădulescu, V., Varga Cs.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge (2010)

  12. Kristály A., Mihăilescu M., Rădulescu V., Tersian S.: Stepan Spectral estimates for a nonhomogeneous difference problem. Commun. Contemp. Math. 12, 1015–1029 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kristály A., Mihăilescu M., Rădulescu V.: Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions. J. Differ. Equ. Appl. 17, 1431–1440 (2011)

    Article  MATH  Google Scholar 

  14. Molica Bisci G., Repovs D.: Nonlinear algebraic systems with discontinuous terms. J. Math. Anal. Appl. 398, 846–856 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Molica Bisci G., Repovs D.: On some variational algebraic problems. Adv. Nonlinear Anal. 2, 127–146 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Molica Bisci G., Repovs D.: On sequences of solutions for discrete anisotropic equations. Expos. Math. 32(3), 284–295 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Molica Bisci G., Repovs D.: Existence of solutions for p-Laplacian discrete equations. Appl. Math. Comput. 242, 454–461 (2014)

    Article  MathSciNet  Google Scholar 

  18. Mihăilescu M., Rădulescu V., Tersian S.: Eigenvalue problems for anisotropic discrete boundary value problems. J. Differ. Equ. Appl. 15, 557–567 (2009)

    Article  MATH  Google Scholar 

  19. Ricceri B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 133, 401–410 (2000)

    Article  MathSciNet  Google Scholar 

  20. Yang Y., Zhang J.: Existence results for a nonlinear system with a parameter. J. Math. Anal. Appl. 340, 658–668 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang Y., Zhang J.: Existence and multiple solutions for a nonlinear system with a parameter. Nonlinear Anal. 70, 2542–2548 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang G.: Existence of non-zero solutions for a nonlinear system with a parameter. Nonlinear Anal. 66(6), 1410–1416 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, G., Bai, L.: Existence of solutions for a nonlinear algebraic system. Discret. Dyn. Nat. Soc. 1–28 (2009)

  24. Zhang G., Cheng S.S.: Existence of solutions for a nonlinear algebraic system with a parameter. J. Math. Anal. Appl. 314, 311–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang G., Feng W.: On the number of positive solutions of a nonlinear algebraic system. Linear Algebr. Appl. 422, 404–421 (2007)

    Article  MATH  Google Scholar 

  26. Zhang G., Liu S.: On a class of semipositone discrete boundary value problems. J. Math. Anal. Appl. 325, 175–182 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Messina, Viale Ferdinando Stagno d’Alcontres, 3198166, Messina, Italy

    Maurizio Imbesi

  2. Dipartimento P.A.U, Università à degli Studi “Mediterranea” di Reggio Calabria, Salita Melissari-Feo di Vito, 89100, Reggio Calabria, Italy

    Giovanni Molica Bisci

Authors
  1. Maurizio Imbesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Molica Bisci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Molica Bisci.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Imbesi, M., Bisci, G.M. Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems. Mediterr. J. Math. 13, 263–278 (2016). https://doi.org/10.1007/s00009-014-0490-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-014-0490-2

Mathematics Subject Classification

Keywords

Search

Navigation