Bifurcation Theory and Related Problems: Anti-Maximum Principle and Resonance1*

Arcoya, David; Gámez, José L.

doi:10.1081/pde-100107462

Cited by 52 publications

(58 citation statements)

References 24 publications

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“…We obtain sufficient conditions for having subcritical (on the left) or supercritical (on the right) bifurcations. Consequently, we also obtain the bifurcation nature of some classical results, such as the LandesmanLazer-type condition for the existence of solutions in the resonant case [15], and the anti-maximum principle [4,10]. We see that the bifurcation point of view also allows us to obtain a local maximum principle for some classes of strongly resonant problems.…”

Section: Introductionmentioning

confidence: 86%

“…⎫ ⎬ ⎭ (6.1) Clement and Peletier [10] and Arcoya and Gámez [4] proved the anti-maximum principle for elliptic problems with Dirichlet boundary condition, where the parameter λ is only in the differential equation in the interior. Arrieta et al .…”

Section: Results Of Anti-maximum-principle Typementioning

confidence: 99%

“…In this paper we analyse the bifurcation induced by nonlinearities in both the differential equation and the boundary condition. The existence of solutions of nonlinear problems when the nonlinearity is in the differential equation and the boundary condition is Dirichlet has been widely studied using (among other techniques) bifurcation theory (see, for example, [4]). However, problems with nonlinear boundary conditions have been less widely studied.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

Mavinga¹,

Pardo²

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider reaction-diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub-or supercritical. We obtain both Landesman-Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

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Section: Introductionmentioning

confidence: 86%

Section: Results Of Anti-maximum-principle Typementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

Mavinga¹,

Pardo²

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…and let us show an Anti-maximum principle for this problem, see [8], [2] for the case where the nonlinear term is in Ω. As usual, we denote by σ 1 the first Steklov eigenvalue and by Φ 1 its positive eigenfunction.…”

Section: The Anti-maximum Principle For the Steklov Problemmentioning

confidence: 99%

“…Also, a form of the anti-maximum principle will also be derived, [8]. A similar analysis for the case of an interior reaction term was first stablished in [2].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Arrieta¹,

Pardo²,

Rodriguez-Bernal³

2007

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

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