种群动力学中的一个不定非线性问题：高多重正解
An indefinite nonlinear problem in population dynamics: high multiplicity of positive solutions
摘要：
反应扩散方程在种群动力学领域有几个应用，其中一些方程的特征在于存在一个因子，该因子描述了异质生境中不同类型的食物源。在这种情况下，研究种群的持续生存或灭绝，它是相关的非平凡的稳定状态的搜索。我们的论文集中在一个一维模型给出的参数依赖方程的形式$u'' + \bigl{(} λa^{+}(t)-μa^{-}(t) \bigr{)}g(u) = 0$，其中$g\colon \mathopen{[}0,1\mathclose{]} \to \mathbb{R}$是一个连续的功能，使$g(0)=g(1)=0$，$g(s) > 0$为每个$0<s<1$和$\lim_{s\to0^{+}}g(s)/s=0$，和重量$a(t)$有两个正驼峰分离的负。以这种方式，我们认为有界的栖息地，其中包括两个有利的食物来源和一个不利的。我们处理各种边界条件，包括Dirichlet和Neumann的，我们证明了八个正解的存在性时，$λ$和$μ$是积极的和充分大的。在整个文件中，利用数值模拟来讨论的结果，并探讨一些开放的问题。
英文摘要：
Reaction-diffusion equations have several applications in the field of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant the search of nontrivial steady states. Our paper focuses on a one-dimensional model given by a parameter-dependent equation of the form $u'' + \bigl{(} λa^{+}(t)-μa^{-}(t) \bigr{)}g(u) = 0$, where $g\colon \mathopen{[}0,1\mathclose{]} \to \mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(s) > 0$ for every $0<s<1$ and $\lim_{s\to0^{+}}g(s)/s=0$, and the weight $a(t)$ has two positive humps separated by a negative one. In this manner, we consider bounded habitats which include two favorable food sources and an unfavorable one. We deal with various boundary conditions, including the Dirichlet and Neumann ones, and we prove the existence of eight positive solutions when $λ$ and $μ$ are positive and sufficiently large. Throughout the paper, numerical simulations are exploited to discuss the results and to explore some open problems.
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