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Boundary eigencurve problems involving the biharmonic operator

Volume 41 / 2014

Omar Chakrone, Najib Tsouli, Mostafa Rahmani, Omar Darhouche Applicationes Mathematicae 41 (2014), 267-275 MSC: Primary 35J35; Secondary 35J40. DOI: 10.4064/am41-2-14

Abstract

The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem $$\left \{ \begin{array}{@{}l@{}} \varDelta^{2}u=\alpha u+\beta\varDelta u \quad \hbox{in}\ \varOmega, \\ u=\varDelta u=0 \quad \hbox{on}\ \partial\varOmega. \end{array} \right.$$ where $(\alpha,\beta)\in\mathbb{R}^{2}$. We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem \begin{equation*} \left\{ \begin{array}{@{}l@{}} \varDelta^2 u=f(u,x)+\beta \varDelta u+h \quad \mbox{in $\varOmega$},\\ \varDelta u=u=0\quad \mbox{on $\partial\varOmega$}, \end{array} \right. \end{equation*} where $f :\varOmega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function and $h$ is a given function in $L^{2}(\varOmega)$.

Authors

  • Omar ChakroneDepartment of Mathematics
    University Mohamed I
    P.O. Box 717
    Oujda 60000, Morocco
    e-mail
  • Najib TsouliDepartment of Mathematics
    University Mohamed I
    P.O. Box 717
    Oujda 60000, Morocco
    e-mail
  • Mostafa RahmaniDepartment of Mathematics
    University Mohamed I
    P.O. Box 717
    Oujda 60000, Morocco
    e-mail
  • Omar DarhoucheDepartment of Mathematics
    University Mohamed I
    P.O. Box 717
    Oujda 60000, Morocco
    e-mail

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