On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion: −du″(x)=g(x)u(x)−u2(x) for x∈R. The unknown function u corresponds to the size of a population. The function g corresponds to the birth (or death) rate of the population which takes on both positive and negative values on R; the −u2 term in the equation corresponds to the fact that the population is self-limiting and the parameter d>0 corresponds to the rate at which the population diffuses. We have obtained our results by the construction of sub and supersolutions and the study of asymptotic properties of solutions. Our results show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady-state distributions of population.