Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media

2006 ◽  
Vol 136 (6) ◽  
pp. 1131-1155 ◽  
Author(s):  
B. Amaziane ◽  
L. Pankratov ◽  
A. Piatnitski
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Equation ◽  
Quasilinear Elliptic Equations ◽  
High Contrast ◽  
Small Measure ◽  
Image Position ◽  
Quasilinear Elliptic

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the form with a high-contrast discontinuous coefficient aε(x), where ε is the parameter characterizing the scale of the microstucture. The coefficient aε(x) is assumed to degenerate everywhere in the domain Ω except in a thin connected microstructure of asymptotically small measure. It is shown that the asymptotical behaviour of the solution uε as ε → 0 is described by a homogenized quasilinear equation with the coefficients calculated by local energetic characteristics of the domain Ω.

Existence of Radial Solutions for Quasilinear Elliptic Equations with Singular Nonlinearities

2003 ◽  
Vol 3 (4) ◽  
Author(s):  
Beatrice Acciaio ◽  
Patrizia Pucci
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Elliptic Equations ◽  
Radial Solutions ◽  
Singular Nonlinearities ◽  
Quasilinear Elliptic

AbstractWe prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f(u) = 0 in ℝ

scholarly journals Unique continuation for non-negative solutions of quasilinear elliptic equations

2001 ◽  
Vol 64 (1) ◽  
pp. 149-156 ◽  
Author(s):  
Pietro Zamboni
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Unique Continuation ◽  
Quasilinear Elliptic Equations ◽  
Kato Class ◽  
Unique Continuation Property ◽  
Continuation Property ◽  
Quasilinear Elliptic

Dedicated to Filippo ChiarenzaThe aim of this note is to prove the unique continuation property for non-negative solutions of the quasilinear elliptic equation We allow the coefficients to belong to a generalized Kato class.

scholarly journals EXISTENCE OF POSITIVE EVANESCENT SOLUTIONS TO SOME QUASILINEAR ELLIPTIC EQUATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 157-162 ◽  
Author(s):  
OCTAVIAN G. MUSTAFA
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Quasilinear Elliptic Equations ◽  
Image Position ◽  
Quasilinear Elliptic

AbstractWe establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as $\vert x\vert \rightarrow +\infty $ under quite general assumptions upon f and g.

scholarly journals A Fredholm alternative for quasilinear elliptic equations with right-hand side measure

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Michele Colturato ◽  
Marco Degiovanni
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Existence Result ◽  
Degree Theory ◽  
Quasilinear Elliptic Equations ◽  
Fredholm Alternative ◽  
Right Hand ◽  
Quasilinear Elliptic

AbstractWe consider a quasilinear elliptic equation with right-hand side measure, which does not satisfy the usual coercivity assumption. We prove an existence result in the line of the Fredholm alternative. For this purpose we develop a variant of degree theory suited to this setting.

scholarly journals On singular quasilinear elliptic equations with data measures

2021 ◽  
Vol 10 (1) ◽  
pp. 1284-1300
Author(s):  
Nour Eddine Alaa ◽  
Fatima Aqel ◽  
Laila Taourirte
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Quasilinear Elliptic Equations ◽  
Main Ingredient ◽  
Uniqueness Results ◽  
Quasilinear Elliptic

Abstract The aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

scholarly journals Multiple Solutions of Quasilinear Elliptic Equations in

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Huei-li Lin
Keyword(s):  
Elliptic Equation ◽  
Ground State ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Quasilinear Elliptic Equation ◽  
Ground State Solution ◽  
The Other ◽  
Quasilinear Elliptic Equations ◽  
Positive Continuous Function ◽  
Quasilinear Elliptic

Assume that is a positive continuous function in and satisfies some suitable conditions. We prove that the quasilinear elliptic equation in admits at least two solutions in (one is a positive ground-state solution and the other is a sign-changing solution).

Quasilinear Elliptic Equations with Singular Nonlinearity

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
João Marcos do Ó ◽  
Esteban da Silva
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Elliptic Equations ◽  
Mems Devices ◽  
Singular Nonlinearity ◽  
Electrostatic Mems ◽  
Quasilinear Elliptic

AbstractIn this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equationAccording to the choice of the parameters α, β, and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the

scholarly journals Multiple solutions of double phase variational problems with variable exponent

2020 ◽  
Vol 13 (4) ◽  
pp. 385-401 ◽  
Author(s):  
Xiayang Shi ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš ◽  
Qihu Zhang
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Quasilinear Equation ◽  
Variational Problems ◽  
Quasilinear Elliptic Equations ◽  
Weighting Method ◽  
Double Phase ◽  
Unbounded Potential ◽  
Quasilinear Elliptic

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like {|\xi|^{q(x)-2}\xi} for small {|\xi|} and like {|\xi|^{p(x)-2}\xi} for large {|\xi|}, where {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)<q(\,\cdot\,)<N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in \mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where {p(\,\cdot\,)} and {q(\,\cdot\,)} are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

scholarly journals Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 128
Author(s):  
Jun Ik Lee ◽  
Yun-Ho Kim
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Functional Method ◽  
Quasilinear Elliptic Equations ◽  
Fountain Theorem ◽  
Small Energy ◽  
Dual Fountain Theorem ◽  
Radially Symmetric Solutions ◽  
Quasilinear Elliptic ◽  
Modified Functional

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the L ∞ -bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose L ∞ -norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools.

scholarly journals On the Harnack inequality for quasilinear elliptic equations with a first-order term

2018 ◽  
Vol 148 (5) ◽  
pp. 1075-1095 ◽  
Author(s):  
Susana Merchán ◽  
Luigi Montoro ◽  
Berardino Sciunzi
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Order Term ◽  
Quasilinear Elliptic Equations ◽  
First Order ◽  
Image Position ◽  
Quasilinear Elliptic ◽  
Comparison Inequality ◽  
Strong Comparison Principle ◽  
Iteration Technique

We consider weak solutions towith p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

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