Existence of three solutions to a double eigenvalue problem for the $p$-biharmonic equation
Volume 104 / 2012
Annales Polonici Mathematici 104 (2012), 71-80
MSC: Primary 35J35; Secondary 58E05.
DOI: 10.4064/ap104-1-5
Abstract
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.
