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∇u∕1+|∇u|2=λa(x)f(u)inΩ,u=0on∂Ω,with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Ω is a bounded domain in RN with a regular boundary ∂Ω, λ>0 represents a diffusivity parameter, a is a continuous weight which may change sign in Ω, 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∈]0,L[. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying λ. Typically, the solutions we find are regular as long as λ is small, while as a consequence of the saturation of the flux they may develop singularities when λ 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.