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

Tellini, Andrea

doi:10.1016/j.jmaa.2018.07.034

Cited by 8 publications

(4 citation statements)

References 22 publications

(36 reference statements)

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“…In addition, similar results to Theorems 1.1 and 1.2 can be established for the Neumann case (1.6). For high multiplicity of positive solutions to a similar type of indefinite superlinear Neumann problem to (1.6), we refer to [31].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

Umezu¹

2021

Preprint

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In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have multiple positive solutions under certain situations. To our end, by conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the unstable positive one. An application of our results to the logistic boundary condition arising from population genetics is given.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

Umezu¹

2021

Preprint

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“…Following a terminology popularized in [31], we refer to (E λ,µ ) as an indefinite equation, meaning that the weight function a(x) changes sign. In the last decades this kind of equations has been widely investigated, both in the ODE and in the PDE setting, starting from the classical contributions [1,2,3,4,13] and till to very recent ones [8,16,27,33,45,46,47]; we refer the reader to [17] for a quite exhaustive bibliography on the subject.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

High multiplicity and chaos for an indefinite problem arising from genetic models

Boscaggin¹,

Feltrin²,

Sovrano³

2019

Preprint

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We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equationwhere λ, µ > 0 are parameters, c ∈ R, a(x) is a locally integrable P -periodic sign-changing weight function, and g : [0, 1] → R is a continuous function such that g(0) = g(1) = 0, g(u) > 0 for all u ∈ ]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 behaviour 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 countable family of globally defined solutions with a complex behaviour, coded by (possibly non-periodic) biinfinite sequences of 3 symbols.

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“…We gather now the main results known for (P α ) in the sublinear case, which are established in [6], [ It is worth pointing out that the uniqueness result in Theorem 1.1(iii) for the Dirichlet and Neumann problems contrasts with some high multiplicity results for positive solutions in the superlinear case [8,33]. In Theorem 1.5(ii) below we shall prove that for q ∈ A N and α > 0 small (P α ) has exactly two positive solutions, which shows that a high multiplicity result does not occur in this situation either.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Kaufmann

Quoirin

Umezu

2020

Annali di Matematica

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Let Ω ⊂ R N (N ≥ 1) be a smooth bounded domain, a ∈ C(Ω) a sign-changing function, and 0 ≤ q < 1. We investigate the Robin problemwhere α ∈ [−∞, ∞) and ν is the unit outward normal to ∂Ω. Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: (Pα) has exactly one nontrivial solution for α ≤ 0, exactly two nontrivial solutions for α > 0 small, and no such solution for α > 0 large. Assuming some further conditions on a, we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work [17], where the cases α = −∞ (Dirichlet) and α = 0 (Neumann) have been considered. We also obtain some results for arbitrary q ∈ [0, 1). Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds. * 2010 Mathematics Subject Classification. 35J15, 35J25, 35J61. †

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

Cited by 8 publications

References 22 publications

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

High multiplicity and chaos for an indefinite problem arising from genetic models

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

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