Picone’s identity for -Laplace operator and its applications

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.

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Pseudo-monotonicity and degenerated or singular elliptic operators

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032184 ◽

1998 ◽

Vol 58 (2) ◽

pp. 213-221 ◽

Cited By ~ 9

Author(s):

P. Drábek ◽

A. Kufner ◽

V. Mustonen

Keyword(s):

Sobolev Spaces ◽

Differential Operators ◽

Weighted Sobolev Spaces ◽

Type Inequality ◽

Elliptic Operators ◽

Hardy Type Inequality ◽

Hardy Type ◽

Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2020-0015 ◽

2020 ◽

Vol 6 (2) ◽

pp. 198-209

Author(s):

Mohamed Laghzal ◽

Abdelouahed El Khalil ◽

My Driss Morchid Alaoui ◽

Abdelfattah Touzani

Keyword(s):

Dirichlet Problem ◽

Variational Approach ◽

Type Inequality ◽

Biharmonic Operator ◽

Nondecreasing Sequence ◽

Hardy Type Inequality ◽

Principal Curve ◽

Hardy Type ◽

Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

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DISCRETE HARDY-TYPE INEQUALITY WITH KERNEL

International Journal of Pure and Apllied Mathematics ◽

10.12732/ijpam.v106i2.11 ◽

2016 ◽

Vol 106 (2) ◽

Author(s):

S. Kumar

Keyword(s):

Type Inequality ◽

Hardy Type Inequality ◽

Hardy Type

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Corrigendum to "Hardy type inequality in variable Lebesgue spaces"

Annales Academiae Scientiarum Fennicae Mathematica ◽

10.5186/aasfm.2010.3542 ◽

2010 ◽

Vol 35 ◽

pp. 679-680 ◽

Cited By ~ 3

Author(s):

Humberto Rafeiro ◽

Stefan Samko

Keyword(s):

Type Inequality ◽

Lebesgue Spaces ◽

Hardy Type Inequality ◽

Variable Lebesgue Spaces ◽

Hardy Type

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Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-033-9 ◽

2010 ◽

Vol 62 (5) ◽

pp. 1116-1130 ◽

Cited By ~ 2

Author(s):

Yongyang Jin ◽

Genkai Zhang

Keyword(s):

Vector Fields ◽

Orthonormal Basis ◽

Type Inequality ◽

Laplacian Operator ◽

Hardy Type Inequality ◽

Invariant Vector ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Left Invariant Vector ◽

Laplacian Operators

AbstractLet 𝔾 be a step-two nilpotent group of H-type with Lie algebra 𝔊 = V ⊕ t. We define a class of vector fields X = {Xj} on 𝔾 depending on a real parameter k ≥ 1, and we consider the corresponding p-Laplacian operator Lp,ku = divX(|∇Xu|p−2∇Xu). For k = 1 the vector fields X = {Xj} are the left invariant vector fields corresponding to an orthonormal basis of V; for 𝔾 being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator Lp,k and as an application, we get a Hardy type inequality associated with X.

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