Boundary eigencurve problems involving the biharmonic operator

Omar Chakrone ¹   ; Najib Tsouli ¹   ; Mostafa Rahmani ¹   ; Omar Darhouche ¹

¹ Department of Mathematics University Mohamed I P.O. Box 717 Oujda 60000, Morocco

Applicationes Mathematicae, Tome 41 (2014) no. 2-3, pp. 267-275

Résumé

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 $L^{2}(\varOmega)$.

Détail

DOI : 10.4064/am41-2-14

Keywords: paper study spectrum fourth order eigenvalue boundary value problem begin array vardelta alpha beta vardelta quad hbox varomega vardelta quad hbox partial varomega end array right where alpha beta mathbb prove existence first nontrivial curve spectrum its variational characterization moreover prove properties curve continuity convexity asymptotic behavior application study non resonance solutions below first principal eigencurve biharmonic problem begin equation* begin array vardelta x beta vardelta quad mbox varomega vardelta quad mbox partial varomega end array right end equation* where varomega times mathbb rightarrow mathbb carath odory function given function varomega