Imperfect bifurcations via topological methods in superlinear indefinite problems

Tellini, Andrea

doi:10.3934/proc.2015.1050

Cited by 3 publications

(4 citation statements)

References 4 publications

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“…The main differences in the results regard the structure of the bifurcation diagrams constructed in this section. Indeed, the secondary bifurcation points on the branches break and give rise to separate connected components (see also [25] for a similar behavior). ◁…”

Section: Which Proves (I)mentioning

confidence: 83%

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

Feltrin¹,

Sovrano²,

Tellini³

2022

DCDS

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We study the second-order boundary value problem<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases}\, -u'' = a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\\, u'(0) = 0, \quad u'(1) = 0,\end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ a_{\lambda,\mu} $\end{document}</tex-math></inline-formula> is a step-wise indefinite weight function, precisely <inline-formula><tex-math id="M2">\begin{document}$ a_{\lambda,\mu}\equiv\lambda $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ [0,\sigma]\cup[1-\sigma,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ a_{\lambda,\mu}\equiv-\mu $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ (\sigma,1-\sigma) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M6">\begin{document}$ \sigma\in\left(0,\frac{1}{2}\right) $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> positive real parameters. We investigate the topological structure of the set of positive solutions which lie in <inline-formula><tex-math id="M9">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> vary. Depending on <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter <inline-formula><tex-math id="M13">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the <inline-formula><tex-math id="M14">\begin{document}$ (\lambda,\mu) $\end{document}</tex-math></inline-formula>-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.

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Section: Which Proves (I)mentioning

confidence: 83%

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

Feltrin¹,

Sovrano²,

Tellini³

2022

DCDS

Self Cite

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“…Moreover, following the steps of [24], it is possible to show that, when c 1 → c 0 the bifurcation diagrams converge, locally uniformly in the complement of the bifurcation points, to the ones of the symmetric case and that, for c 0 ∼ c 1 , imperfect bifurcations occur. French National Research Agency under the project "NONLOCAL"(ANR-14-CE25-0013), and by the Spanish Ministry of Economy, Industry and Competitiveness through project MTM2015-65899-P and contract Juan de la Cierva Incorporación IJCI-2015-25084.…”

Section: Bifurcation Diagrams In α and General Multiplicity Resultsmentioning

confidence: 99%

“…Remark 5.2. It is possible to adapt the analysis of [18,24] to study the case in which the weight function a(t) in (1.1) is asymmetric of the form…”

Section: Bifurcation Diagrams In α and General Multiplicity Resultsmentioning

confidence: 99%

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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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Global structure of the set of 1-node solutions in a class of degenerate diffusive logistic equations

Cubillos

López-Gómez

Tellini

2023

Communications in Nonlinear Science and Numerical Simulation

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Imperfect bifurcations via topological methods in superlinear indefinite problems

Cited by 3 publications

References 4 publications

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

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

Global structure of the set of 1-node solutions in a class of degenerate diffusive logistic equations

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