Existence of three solutions for a 2-dimensional Navier Problem

Abstract

In this paper, the existence of at least three weak solutions for Navier problem \begin{equation*}\begin{cases}\Delta\left(|\Delta u|\Delta u\right)=\lambda f(x,u) + \mu g(x,u), & \text{in}\, \Omega, \\u=\Delta u=0, & \text{on}\, \partial \Omega,\end{cases}\end{equation*}where $\Omega \subset \mathbb{R}^2$ is non-empty bounded open set with smooth boundary $\partial \Omega$, $\lambda$, $\mu \in [0,+\infty)$ and $f$, $g : \Omega \times \mathbb{R} \to \mathbb{R}$ are $L^1$-Carath\'eodory functions, is established. The approach is based on variational methods and critical points.