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
2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato

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.


2014 ◽  
Vol 71 (2) ◽  
pp. 141-156 ◽  
Author(s):  
Pavel Exner ◽  
Alexander Minakov ◽  
Leonid Parnovski

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Calogero Vetro

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.


2017 ◽  
Vol 22 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Alexey V. Filinovskiy

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.


2010 ◽  
Vol 12 (06) ◽  
pp. 953-975 ◽  
Author(s):  
J. CHABROWSKI ◽  
I. PERAL ◽  
B. RUF

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.


2013 ◽  
Vol 13 (1) ◽  
Author(s):  
Daniel Daners

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.


2014 ◽  
Vol 139 (2) ◽  
pp. 341-352 ◽  
Author(s):  
Alexey Filinovskiy

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