Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

In this paper we consider the equation \begin{document}$(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$\end{document} with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and \begin{document}$f, g$\end{document} are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form \begin{document}$λ>\overline λ$\end{document} and \begin{document}$0 , then we provide further conditions under which, respectively, the bound \begin{document}$\overlineμ(λ)$\end{document} is either necessary, or can be removed.