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On the spectrum of the $p$-biharmonic operator involving $p$-Hardy's inequality

Volume 41 / 2014

Abdelouahed El Khalil, My Driss Morchid Alaoui, Abdelfattah Touzani Applicationes Mathematicae 41 (2014), 239-246 MSC: Primary 35J35; Secondary 35J40. DOI: 10.4064/am41-2-11

Abstract

In this paper, we study the spectrum for the following eigenvalue problem with the $p$-biharmonic operator involving the Hardy term: $$\varDelta (|\varDelta u|^{p-2}\varDelta u)= \lambda \frac {|u|^{p-2}u}{\delta (x)^{2p}} \hbox { in } \varOmega , \ u\in W_0^{2,p}(\varOmega ).$$ By using the variational technique and the Hardy–Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

Authors

  • Abdelouahed El KhalilDepartment of Mathematics and Statistics
    College of Science
    Al-Imam Mohammad Ibn Saud
    Islamic University (IMSIU)
    P.O. Box 90950
    Riyadh 11623, Saudi Arabia
    e-mail
  • My Driss Morchid AlaouiFaculty of Sciences Dhar-Mahraz
    Department of Mathematics
    University Sidi Mohamed Ben Abdellah
    P.O. Box 1796 Atlas
    Fez 30000, Morocco
    e-mail
  • Abdelfattah TouzaniFaculty of Sciences Dhar-Mahraz
    Department of Mathematics
    University Sidi Mohamed Ben Abdellah
    P.O. Box 1796 Atlas
    Fez 30000, Morocco
    e-mail

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