Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros (Q1661142)

The following problem is considered. \[ \begin{cases} (-\Delta)^k u = \lambda f(x,u) + \mu g(x,u) & \text{in} \ \Omega,\\ u>0 & \text{in} \ \Omega, \\ (-\Delta)^i u = 0 & \text{on} \ \partial\Omega, \ i=0,\dots,k-1 \end{cases},\] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $\lambda$ and $\mu$ are nonnegative parameters, $f$ and $g$ are nonnegative functions and $k \in\mathbb{N}$. The regions of two parameters $(\lambda,\mu)$ for the existence, nonexistence and multiplicity of positive solutions are obtained.