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Existence of Two Non-zero Weak Solutions for a Nonlinear Navier Boundary Value Problem Involving the \(p\)-Biharmonic

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Abstract

We study the existence of two non-zero solutions for a class of elliptic Navier boundary value problems in which the \(p\)-biharmonic operator arises. The approach is based on variational methods.

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References

  1. Afrouzi, G., Mirzapour, M., Chung, N.T.: Existence and non-existence of solutions for a \(p(x)\)-biharmonic problem. Electron. J. Differ. Equ. 2015(158), 1 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Bonanno, G.: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75, 2992–3007 (2012)

    Article  MathSciNet  Google Scholar 

  3. Bonanno, G., Chinnì, A., Tersian, S.A.: Existence results for a two point boundary value problem involving a fourth-order equation. Electron. J. Qual. Theory Differ. Equ. 33, 1-9 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Bonanno, G., D’Aguì, G.: Two non-zero solutions for elliptic Dirichlet problems. Z. Anal. Anwend. 35(4), 449–464 (2016)

    Article  MathSciNet  Google Scholar 

  5. Candito, P., Li, L., Livrea, R.: Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the \(p\)-biharmonic. Nonlinear Anal. 75, 6360–6369 (2012)

    Article  MathSciNet  Google Scholar 

  6. Ding, L.: Multiple solutions for a perturbed Navier boundary value problem involving the p-biharmonic. Bull. Iran. Math. Soc. 41(1), 269–280 (2015)

    MathSciNet  MATH  Google Scholar 

  7. El Amrouss, A., Ourraoui, A.: Existence of solutions for a boundary problem involving p(x)-biharmonic operator. Bol. Soc. Parana. Mat. 31(1), 179–192 (2013)

    Article  MathSciNet  Google Scholar 

  8. Kong, L.: Multiple solutions for fourth order elliptic problems with \(p(x)\)-biharmonic operators. Opusc. Math. 36(2), 253–264 (2016)

    Article  MathSciNet  Google Scholar 

  9. Li, L.: Existence of three solutions for p-biharmonic equation. Int. J. Nonlinear Anal. Appl. 2011, 1–9 (2011)

    Google Scholar 

  10. Li, L.: Existence of three solutions for a class of Navier quasilinear elliptic systems involving the \((p_{1},ldots, p_{n})\)-biharmonic. Bull. Korean Math. Soc. 50(1), 57–71 (2013)

    Article  MathSciNet  Google Scholar 

  11. Li, L., Ding, L., Pan, W.: Existence of multiple solutions for a \(p(x)\)-biharmonic equation. Electron. J. Differ. Equ. 2013(139), 1 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Li, C., Tang, C.: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 72, 1339–1347 (2010)

    Article  MathSciNet  Google Scholar 

  13. Li, L., Tang, C.: Existence of three solutions for (p, q)-biharmonic systems. Nonlinear Anal. 73, 796–805 (2010)

    Article  MathSciNet  Google Scholar 

  14. Massar, M., El Hssini, M., Tsouli, N.: Existence and multiplicity of solutions for class of Navier boundary p-biharmonic problem near resonance. Bol. Soc. Parana. Mat. 32(2), 83–93 (2014)

    Article  MathSciNet  Google Scholar 

  15. Massar, M., Rahmani, E., Tsouli, N.: Existence of three solutions for a biharmonic system with weight. Adv. Dyn. Syst. Appl. 8(1), 95–106 (2013)

    MathSciNet  Google Scholar 

  16. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Conference Board of the Mathematical Sciences, vol. 65. Am. Math. Soc., Providence (1986)

    Book  Google Scholar 

  17. Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 3(4), 697–718 (1976)

    MathSciNet  MATH  Google Scholar 

  18. Tang, X., Zhang, J.: Multiple results to some biharmonic problems. Abstr. Appl. Anal. 2014, 1 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Yin, H., Liu, Y.: Existence of three solutions for a Navier boundary value problem involving the \(p(x)\)- biharmonic. Bull. Korean Math. Soc. 50, 1817–1826 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first two authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. Department of Engineering, University of Messina, C.da Di Dio, Sant’Agata, 98166, Messina, Italy

    Gabriele Bonanno & Antonia Chinnì

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Gabriele Bonanno
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  2. Antonia Chinnì
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  3. Donal O’Regan
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Correspondence to Gabriele Bonanno.

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Bonanno, G., Chinnì, A. & O’Regan, D. Existence of Two Non-zero Weak Solutions for a Nonlinear Navier Boundary Value Problem Involving the \(p\)-Biharmonic. Acta Appl Math 166, 1–10 (2020). https://doi.org/10.1007/s10440-019-00251-7

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  • DOI: https://doi.org/10.1007/s10440-019-00251-7

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