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 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.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

2018 ◽  
Vol 61 (4) ◽  
pp. 943-959 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Reaction Function ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Potential Term ◽  
Symmetric Mountain Pass Theorem ◽  
Indefinite Potential ◽  
Concave Term

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

Positive solutions of nth-order impulsive eigenvalue problems with an advanced argument

2017 ◽  
Vol 10 (04) ◽  
pp. 1750072
Author(s):  
Gaoli Lu ◽  
Meiqiang Feng
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Hölder’S Inequality ◽  
Transformation Technique ◽  
Hölder's Inequality ◽  
Advanced Argument

In this paper, we study a [Formula: see text]th-order impulsive eigenvalue problem with an advanced argument. We shall establish several criteria for the optimal intervals of the parameter [Formula: see text] so as to ensure existence of single or many positive solutions. Our methods are based on transformation technique, Hölder’s inequality and the eigenvalue theory.

scholarly journals Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

2017 ◽  
Vol 37 (5) ◽  
pp. 2589-2618 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš ◽  
◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Indefinite Potential

Linked eigenvalue problems for the p-Laplacian

1994 ◽  
Vol 124 (5) ◽  
pp. 1023-1036
Author(s):  
P. A. Binding ◽  
Y. X. Huang
Keyword(s):  
Existence And Uniqueness ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Smooth Domain ◽  
Uniqueness Of Solutions ◽  
Oscillation Theorem ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

Linked equations of the formare studied on a bounded smooth domain in RN for λ ∈ R2. Existence and uniqueness of solutions are discussed for fi homogeneous of order p – 1 in ui, generalising the ‘Klein Oscillation Theorem’ when p = 2, N = 1. Bifurcation from the principal eigenvalue is also considered for nonhomogeneous perturbations fi of order greater than p – 1.

scholarly journals Positive solutions for nonlinear nonhomogeneous parametric Robin problems

2018 ◽  
Vol 30 (3) ◽  
pp. 553-580 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Differential Operator ◽  
Positive Solution ◽  
Positive Solutions ◽  
Type Theorem ◽  
Sobolev Norm ◽  
Robin Problem ◽  
Reaction Term ◽  
Nonhomogeneous Differential Operator ◽  
Nonlinear Nonhomogeneous Differential Operator

AbstractWe study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter {\lambda>0} approaches zero, we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally, we show that for every admissible parameter value, there is a smallest positive solution {u^{*}_{\lambda}} of the problem, and we investigate the properties of the map {\lambda\mapsto u^{*}_{\lambda}}.

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.

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