Existence of three solutions to a double eigenvalue problem for the $p$-biharmonic equation

Lin Li¹ ; Shapour Heidarkhani²

¹ Department of Science Sichuan University of Science and Engineering 643000 Zigong, P.R. China
² Department of Mathematics Faculty of Sciences Razi University 67149 Kermanshah, Iran

Annales Polonici Mathematici, Tome 104 (2012) no. 1, pp. 71-80

Résumé

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem $$ \begin{cases} \varDelta(|\varDelta u|^{p-2}\varDelta u) - \text{div} (|\nabla u|^{p-2} \nabla u)=\lambda f(x,u) + \mu g(x,u) \hbox{in }\varOmega,\\ u=\varDelta u=0 \hbox{on }\partial \varOmega,\end{cases} $$ where $\varOmega \subset \mathbb{R}^N$ $(N \geq 1)$ is a non-empty bounded open set with a sufficiently smooth boundary $\partial \varOmega$, $\lambda>0$, $\mu>0$ and $f,g : \varOmega \times \mathbb{R} \to \mathbb{R}$ are two $L^1$-Carathéodory functions.

Détail

DOI : 10.4064/ap104-1-5

Keywords: using three critical points theorem variational methods study existence least three weak solutions navier problem begin cases vardelta vardelta p vardelta text div nabla p nabla lambda hbox varomega vardelta hbox partial varomega end cases where varomega subset mathbb geq non empty bounded set sufficiently smooth boundary partial varomega lambda varomega times mathbb mathbb carath odory functions