New exact multiplicity results with an application to a population model

We obtain some new exact multiplicity results for the Dirichlet boundary-value problem Deltau + gimelf(u) = 0 for x is an element of B-n, u = 0 for x is an element of partial derivativeB(n) on a unit ball B-n in 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.