Multiple positive solutions for a superlinear problem: A topological approach

Feltrin, Guglielmo; Zanolin, Fabio

doi:10.1016/j.jde.2015.02.032

Cited by 36 publications

(66 citation statements)

References 32 publications

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“…We conclude this section with a further open problem motivated by the papers [6,19,20], where g(u) ∼ u p , p > 1, or the paper [9], where g(u) ∼ u 2 /(1 + u 2 ). In these works, the weight term has m intervals of positivity separated by intervals of negativity which characterize the number of positive solutions.…”

Section: Discussion: Numerical Examples and Future Perspectivesmentioning

confidence: 99%

An indefinite nonlinear problem in population dynamics: high multiplicity of positive solutions

Feltrin

Sovrano

2018

Nonlinearity

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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).

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Section: Discussion: Numerical Examples and Future Perspectivesmentioning

confidence: 99%

An indefinite nonlinear problem in population dynamics: high multiplicity of positive solutions

Feltrin

Sovrano

2018

Nonlinearity

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“…This kind of multiplicity was then proved when λ = 0 and the negative part of the weight is sufficiently large in [12], by using a shooting technique (we also mention that the same results have been obtained for large solutions in [8]), and, later, in [5], in the PDE case by means of variational methods. Recently, in [10], the use of topological degree has allowed the authors to obtain the same kind of results for more general nonlinearities and λ ∼ 0.…”

Section: Introductionmentioning

confidence: 77%

High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions

Tellini

2018

Journal of Mathematical Analysis and Applications

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We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The sign-changing weight in front of the nonlinearity is taken to be piecewise constant, which allows us to perform a sharp phase-plane analysis, firstly to study the sets of points reached at the end of the regions where the weight is negative, and then to connect such sets through the flow in the positive part. Moreover, we study how the number of solutions depends on the amplitude of the region in which the weight is positive, using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams.

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“…In , the authors study the equation

\{\begin{matrix} z^{′′} + a (t) f (z (t)) = 0, t \in [0, L], \\ z (0) = z (L) = 0, \end{matrix}

where f (0) = 0 and f satisfies (H1) and (H2) .…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In [11], the authors study the equation The big task to deal directly with the differential equations is the difficulty to obtain explicitly the solution of the problem or even to understand the behavior of the solution to be able to study the important properties such as stability, bifurcation, and existence of the attractors, among others. In the real world, the situations where an exact solution exists are very few.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%