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Explicit \(L^{\infty }\)-norm estimates via Morse index for the bi-harmonic and tri-harmonic semilinear problems

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Abstract

In this paper, we establish \(L^{\infty }\) and \(L^{p}\) estimates for solutions of some polyharmonic elliptic equations via the Morse index. As far as we know, it seems to be the first time that such explicit estimates are obtained for polyharmonic problems.

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References

  1. Bahri, A., Lions, P.L.: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. XLV, 1205–1215 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang, K.C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)

    Book  MATH  Google Scholar 

  3. Hajlaoui, H., Harrabi, A., Mtiri, F.: Morse indices of solutions for super-linear elliptic PDEs. Nonlinear Anal. 116, 180–192 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Recheil, W., Weth, T.: A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems. Math. Z. 261, 805–827 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sirakov, B.: Existence results and a priori bounds for higher order elliptic equations and systems. J. Math. Pures Appl. 89, 114–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Soranzo, R.: A priori estimates and existence of positive solutions of a superlinear polyharmonic equation. Dyn. Syst. Appl. 3, 465–487 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Schechter, M., Zou, W.: Critical Point Theory and Its Applications. Springer, New York (2006)

    MATH  Google Scholar 

  8. Yang, X.: Nodal sets and Morse indices of solutions of super-linear elliptic PDEs. J. Funct. Anal. 160, 223–253 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. ANLIG, UR13ES32, University of Tunis El-Manar, 2092, El Manar II, Tunisia

    Foued Mtiri

  2. Department of Mathematics, Northern Border University, Arar, Saudi Arabia

    Abdellaziz Harrabi

  3. Institut de Mathématiques Appliquées et de l’Informatique, Université de Kairouan, Kairouan, Tunisia

    Abdellaziz Harrabi

  4. IECL, UMR 7502, Université de Lorraine, 3 rue Augustin Fresnel, 57073, Metz, France

    Dong Ye

Authors
  1. Foued Mtiri
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  2. Abdellaziz Harrabi
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  3. Dong Ye
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Correspondence to Foued Mtiri.

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Mtiri, F., Harrabi, A. & Ye, D. Explicit \(L^{\infty }\)-norm estimates via Morse index for the bi-harmonic and tri-harmonic semilinear problems. manuscripta math. 159, 57–79 (2019). https://doi.org/10.1007/s00229-018-1037-9

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  • DOI: https://doi.org/10.1007/s00229-018-1037-9

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