Boundary eigencurve problems involving the biharmonic operator
Volume 41 / 2014
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 $L^{2}(\varOmega)$.
