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A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

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Abstract

We consider here solutions of a nonlinear Neumann elliptic equation Δu + f (x, u) = 0 in Ω, ∂u/∂ν = 0 on ∂Ω, where Ω is a bounded open smooth domain in \({\mathbb{R}^N, N\geq2}\) and f satisfies super-linear and subcritical growth conditions. We prove that L ∞−bounds on solutions are equivalent to bounds on their Morse indices.

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References

  1. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahri A., Lions P.L.: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure. App. Math. 45, 1205–1215 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Courant R., Hilbert D.: Methods of mathematical Physics, vol. I. Interscience, NY (1953)

    Google Scholar 

  4. Dancer E.N.: Finite Morse index solutions of supercritical problems. J. Reine Angew. Math. 620, 213–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1. Springer-Verlag

  6. de Figueiredo D.G., Yang J.: On a semilinear elliptic problem without (PS) condition. J. Diff. Equat. 187, 412–428 (2003)

    Article  MATH  Google Scholar 

  7. Gidas B., Spruck J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Part. Differ. Equat. 6, 883–901 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilbarg D., Trudinger N.S.: Elliptic Partial Differntial Equations of Second Order, Grundlehren Math. Wiss, vol. 224. Springer-Verlag, New York (1977)

    Google Scholar 

  9. Harrabi A., Rebhi S., Selmi A.: Solutions of superlinear equations and their Morse indices, I. Duke Math. J. 94, 141–157 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harrabi A., Rebhi S., Selmi A.: Solutions of superlinear equations and their Morse indices, II. Duke Math. J. 94, 159–179 (1998)

    Article  MathSciNet  Google Scholar 

  11. Wang G., Ye D.: On a nonlinear elliptic equation arising in a free boundary problem. Math. Z. 244, 531–548 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Weyl H.: das asymptotische Verteilungsgesetz der eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1911)

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Département de Mathématiques, Institue de Mathématiques Appliquées et d’Informatiques de Kairouan, Kairouan, Tunisia

    Abdellaziz Harrabi

  2. Tübingen University, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Mohameden Ould Ahmedou

  3. Département de Mathématiques, Faculté des Sciences deTunis, University Elmanar, Campus Universitaire, 2092, Tunis, Tunisia

    Salem Rebhi

  4. Département de Mathématiques, Faculté des Sciencesde Bizerte, Zarzouna, 7021, Bizerte, Tunisia

    Abdelbaki Selmi

Authors
  1. Abdellaziz Harrabi
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  2. Mohameden Ould Ahmedou
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  3. Salem Rebhi
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  4. Abdelbaki Selmi
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Correspondence to Mohameden Ould Ahmedou.

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Harrabi, A., Ahmedou, M.O., Rebhi, S. et al. A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case. manuscripta math. 137, 525–544 (2012). https://doi.org/10.1007/s00229-011-0488-z

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  • DOI: https://doi.org/10.1007/s00229-011-0488-z

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