Existence of positive solutions for a class of Berger's equations

In this paper we study the existence of positive solutions of Berger's equation with the Navier boundary condition
\begin{equation*} \begin{cases} \Delta^2 u-\biggl(a+b\displaystyle\int_\Omega |\nabla u|^2\biggr)\Delta u=\lambda f(u), & x\in \Omega, \\ u=\Delta u=0, &x\in \partial\Omega, \end{cases} \end{equation*}
where $\lambda$ is a positive parameter, $f\in C(\mathbb{R},\mathbb{R})$ is a given function, $a$ and $b$ are two constants with $b\geq 0$, and $a+bt$ is allowed to be negative for some $t\geq0$. The proof of the main results is based on the topological degree theory.