Positive solutions of indefinite logistic growth models with flux-saturated diffusion

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator - div (del u/root 1+vertical bar del u vertical bar(2)) - lambda a(x)f(u) in Omega, u - 0 on partial derivative Omega, with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Omega is a bounded domain in R-N with a regular boundary partial derivative Omega, lambda > 0 represents a diffusivity parameter, a is a continuous weight which may change sign in Omega, and f: [0, L] -> R, with L > 0 a given constant, is a continuous function satisfying f(0) = f (L) = 0 and f (s) > 0 for every s is an element of [0, L]. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying lambda. Typically, the solutions we find are regular as long as lambda is small, while as a consequence of the saturation of the flux they may develop singularities when A becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f (s) = s(L - s) and a 1, having no similarity with the case of linear diffusion based on the Fick-Fourier's law