New exact multiplicity results with an application to a population model (Q2771447)

We obtain some new exact multiplicity results for the Dirichlet boundary-value problem \(\Delta u+\lambda f(u)=0\) for \(x\in \mathbb{B}^n\), \(u=0\), for \(x\in\delta \mathbb{B}^n\) on a unit ball \(\mathbb{B}^n\) in \(\mathbb{R}^n\). We consider several classes of nonlinearities \(f(u)\), including both positive and sign-changing cases. A crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem. As an application we obtain exact multiplicity results for the Holling-Tanner population model.