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On the eigencurves of one dimensional p-Laplacian with weights for an elliptic Neumann problem

  • Ahmed SanhajiEmail author
  • Ahmed Dakkak
Article
  • 5 Downloads

Abstract

In the present paper we study the existence of the eigencurves of one dimensional p-Laplacian with indefinite weights of the Neumann problem for the following elliptic equation
$$\begin{aligned} \left\{ \begin{array}{ll} -\left( |u'|^{p-2}u'\right) ' =\left( \alpha \, m_{1}(x)+\beta \,m_{2}(x)\right) |u|^{p-2}u\quad \text{ in } \; I = {]a,b[}\\ \quad \\ u'(a)=u'(b) = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{array} \right. \end{aligned}$$
Moreover, we establish their variational formulations and asymptotic behavior. Finally, we prove some properties of the principal eigencurve such as concavity and differentiability.

Keywords

p-Laplacian Eigencurves One dimensional 

Mathematics Subject Classification

35J30 35J60 35J66 

Notes

References

  1. 1.
    Anane, A.: Simplicit et isolation de la premire valeur propre du p-Laplacien avec poids. C.R.A.S Paris t 305, 725–728 (1987)zbMATHGoogle Scholar
  2. 2.
    Anane, A., Chakrone, O., Moussa, M.: Spectrum of one dimensional p-laplacian operator with indefinite weight. EJQTDE 17, 1–11 (2002)zbMATHGoogle Scholar
  3. 3.
    Binding, P.A., Huang, Y.X.: Bifurcation from eigencurves of the p-Laplacian. Differ. Integral Equ. 8, 405–414 (1995)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dakkak, A., Anane, A.: Nonresonance conditions on the potential for a Neumann problem. In: Proceedings of the International Conference Held in Fez, Partial Differential Equations, pp. 85–102 (2001)Google Scholar
  5. 5.
    Dakkak, A., El Habib, S., Tsouli, N.: On the spectrum of one dimensional p-Laplacian for an eigenvalue problem with Neumann boundary conditions. Bol. Soc. Paran de Mat. (3s.) v. 33(1), 9–21 (2015)Google Scholar
  6. 6.
    Dakkak, A., Hadda, M.: Eigencurves of the p-Laplacian with weights and their asymptotic behavior. Electron. J. Differ. Equ. 35, 1–7 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dakkak, A., Moussaoui, M.: On the second eigencurve for the p-laplacian operator with weight. Bol. Soc. Paran de Mat. (3s.) v. 35(1), 281–289 (2017)Google Scholar
  8. 8.
    Godoy, T., Gossez, J.-P., Paczka, S.: On the antimaximum principle for the p-Laplacian with indefinite weight. Nonlinear Anal. 51, 449–467 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guedda, M., Veron, L.: Bifurcation phenomena associated to the p-Laplace operator. Trans. Am. Math. Soc. 310, 419–431 (1988)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Huang, Y.X.: On eigenvalue problems of the p-Laplacian with Neumann boundary conditions. Proc. Am. Math. Soc. 109(1), 177–184 (1990)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sanhaji, A., Dakkak, A.: Nonresonance Conditions on the Potential for a Nonlinear Nonautonomous Neumann Problem. Bol. Soc. Paran de Mat.  https://doi.org/10.5269/bspm.v38i3.36296

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory LSI Department of Mathematics, Polydisciplinary, Faculty of TAZAUniversity Sidi Mohamed Ben AbdellahTAZAMorocco

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