We deal with the Neumann boundary value problemwhere the weight term has two positive humps separated by a negative one and g : [0, 1] → R is a continuous function such that g(0) = g(1) = 0, g(s) > 0 for 0 < s < 1 and lim s→0 + g(s)/s = 0. We prove the existence of three solutions when λ and µ are positive and sufficiently large.

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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 + λa + (t) − µa − (t) g(u) = 0, where g : [0, 1] → R is a continuous function such that g(0) = g(1) = 0, g(s) > 0 for every 0 < s < 1 and lim s→0 + 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. * Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2017: "Problemi differenziali con peso indefinito: tra metodi topologici e aspetti dinamici". Guglielmo Feltrin is supported by the Belgian F.R.S.-FNRS -Fonds de la Recherche Scientifique, Chargé de recherches project: "Quantitative and qualitative properties of positive solutions to indefinite problems arising from population genetic models: topological methods and numerical analysis", and partially by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14).

In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation{x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s}, wheresis a real parameter,fandgare continuous functions andgisT-periodic in the variablet. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions ong. Analogous results are also obtained for Neumann boundary conditions.

We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary migration, that is independent of genotype, in absence of mutation and random drift. The focus is on a conjecture, that was raised up in literature of population genetics, about the possible uniqueness of polymorphic equilibria, which are known as clines, under particular circumstances. We study the number of these equilibria, making use of topological tools, and we give a negative answer to that question by means of two examples. Indeed, we provide numerical evidence of multiplicity of positive solutions for two different Neumann problems satisfying the requests of the conjecture.

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We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equationu^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0,where {\lambda,\mu>0} are parameters, {c\in\mathbb{R}}, {a(x)} is a locally integrable P-periodic sign-changing weight function, and {g\colon{[0,1]}\to\mathbb{R}} is a continuous function such that {g(0)=g(1)=0}, {g(u)>0} for all {u\in{]0,1[}}, with superlinear growth at zero. A typical example for {g(u)}, that is of interest in population genetics, is the logistic-type nonlinearity {g(u)=u^{2}(1-u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of {a(x)}. More precisely, when m is the number of intervals of positivity of {a(x)} in a P-periodicity interval, we prove the existence of {3^{m}-1} non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.

This paper analyzes the superlinear indefinite prescribed mean curvature problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with a regular boundary [Formula: see text], [Formula: see text] satisfies [Formula: see text], as [Formula: see text], [Formula: see text] being an exponent with [Formula: see text] if [Formula: see text], [Formula: see text] represents a parameter, and [Formula: see text] is a sign-changing function. The main result establishes the existence of positive regular solutions when [Formula: see text] is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for [Formula: see text] small is further discussed assuming that [Formula: see text] satisfies [Formula: see text] as [Formula: see text], [Formula: see text] being such that [Formula: see text] if [Formula: see text]; thus, in dimension [Formula: see text], the function [Formula: see text] is not superlinear at [Formula: see text], although its potential [Formula: see text] is. Imposing such different degrees of homogeneity of [Formula: see text] at [Formula: see text] and at [Formula: see text] is dictated by the specific features of the mean curvature operator.