scholarly journals On the principal eigenvalue of a Robin problem with a large parameter

2008 ◽  
Vol 281 (2) ◽  
pp. 272-281 ◽  
Author(s):  
Michael Levitin ◽  
Leonid Parnovski
Keyword(s):  
Principal Eigenvalue ◽  
Robin Problem ◽  
Large Parameter

scholarly journals Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Potential Term ◽  
Indefinite Potential

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When $$q=2$$ q = 2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.

scholarly journals Asymptotic eigenvalue estimates for a Robin problem with a large parameter

2014 ◽  
Vol 71 (2) ◽  
pp. 141-156 ◽  
Author(s):  
Pavel Exner ◽  
Alexander Minakov ◽  
Leonid Parnovski
Keyword(s):  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalue Estimates ◽  
Asymptotic Eigenvalue

scholarly journals Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Calogero Vetro
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Admissible Range ◽  
Indefinite Potential ◽  
Multiplicity Of Positive Solutions

AbstractWe consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation $$f(z,\cdot )$$ f ( z , · ) is $$(p-1)$$ ( p - 1 ) -sublinear and then the case where it is $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter $$\lambda \in {\mathbb {R}}$$ λ ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS OF THE ROBIN PROBLEM WITH LARGE PARAMETER

2017 ◽  
Vol 22 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalues And Eigenfunctions ◽  
The Difference ◽  
Robin Boundary ◽  
Outward Unit

We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn , n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem for α → +∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions.

scholarly journals ON AN EIGENVALUE PROBLEM INVOLVING THE HARDY POTENTIAL

2010 ◽  
Vol 12 (06) ◽  
pp. 953-975 ◽  
Author(s):  
J. CHABROWSKI ◽  
I. PERAL ◽  
B. RUF
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Principal Eigenvalue ◽  
Robin Boundary Conditions ◽  
Robin Problem ◽  
Hardy Potential ◽  
High Dimensions ◽  
The Neumann Problem ◽  
Robin Boundary ◽  
Second Eigenvalue

In this note we consider the eigenvalue problem for the Laplacian with the Neumann and Robin boundary conditions involving the Hardy potential. We prove the existence of eigenfunctions of the second eigenvalue for the Neumann problem and of the principal eigenvalue for the Robin problem in "high" dimensions.

Eigenvalue Problems for a Cooperative System with a Large Parameter

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
Daniel Daners
Keyword(s):  
Eigenvalue Problems ◽  
Elliptic Boundary ◽  
Principal Eigenvalue ◽  
Limit Problem ◽  
Bilinear Forms ◽  
Cooperative System ◽  
Large Parameter ◽  
New Approach ◽  
Elliptic Boundary Value ◽  
The Resolvent Operator

AbstractWe consider the principal eigenvalue of a cooperative system of elliptic boundary value problems as a parameter tends to infinity. The main aim is to introduce a new approach to deal with the limit problem by focusing on the resolvent operator corresponding to the system rather than the eigenvalue problem itself. This allows the consistent use of elementary properties of bilinear forms and the semi-groups they induce. At the same time we weaken assumptions in related work.

scholarly journals On the eigenvalues of a Robin problem with a large parameter

2014 ◽  
Vol 139 (2) ◽  
pp. 341-352 ◽  
Author(s):  
Alexey Filinovskiy
Keyword(s):  
Robin Problem ◽  
Large Parameter
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