This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator $-\mathrm{div}\left(\nabla u∕\sqrt{1+{|\nabla u|}^2}\right)=\lambda a\left(x\right)f\left(u\right)\text{in}\Omega ,u=0\text{on}\partial \Omega ,$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 ${\mathbb{R}}^N$ with a regular boundary $\partial \Omega$, $\lambda >0$ represents a diffusivity parameter, $a$ is a continuous weight which may change sign in $\Omega$, and $f:[0,L]\to \mathbb{R}$, with $L>0$ a given constant, is a continuous function satisfying $f\left(0\right)=f\left(L\right)=0$ and $f\left(s\right)>0$ for every $s\in ]0,L[$. 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 $\lambda$ becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, $f\left(s\right)=s(L-s)$ and $a\equiv 1$, having no similarity with the case of linear diffusion based on the Fick–Fourier’s law.

本文分析了由平均曲率算子驱动的拟线性椭圆边值问题 $-\mathrm{div}\left(\nabla ü∕\sqrt{\mathrm{1个}+{|\nabla ü|}^2}\right)=\lambda \mathrm{一种}（X）F（ü）\text{在}\Omega ，ü=0\text{上}\partial \Omega ，$目的是了解在具有空间异质性的逻辑增长模型中通量饱和扩散的影响。这里，$\Omega$ 是一个有界域 ${\mathbb{[R}}^{ñ}$ 有规则的边界 $\partial \Omega$， $\lambda >0$ 代表扩散系数， $\mathrm{一种}$ 是连续的重量，可能会更改登录 $\Omega$和 $F：[0，\mathrm{大号}]\to \mathbb{[R}$，带有 $\mathrm{大号}>0$ 给定常数是满足以下条件的连续函数 $F（0）=F（\mathrm{大号}）=0$ 和 $F（s）>0$ 每一个 $s\in ]0，\mathrm{大号}[$。取决于行为$F$ 零时，通过改变三个定性分叉图出现 $\lambda$。通常情况下，只要$\lambda$ 很小，而由于通量饱和，当通量饱和时，它们可能会产生奇点 $\lambda$变得更大。即使对于最简单的逻辑模型，也可以检测到相当意外的多重性现象，$F（s）=s（\mathrm{大号}-s）$ 和 $\mathrm{一种}\equiv \mathrm{1个}$，与基于Fick-Fourier定律的线性扩散情况没有相似之处。