scholarly journals Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

2003 ◽  
Vol 2003 (12) ◽  
pp. 715-741 ◽  
Author(s):  
Imed Bachar ◽  
Habib Màagli ◽  
Syrine Masmoudi ◽  
Malek Zribi
Keyword(s):  
Green Function ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Singular Solutions ◽  
Kato Class ◽  
New Class ◽  
Nonlinear Elliptic ◽  
New Form ◽  
The Green Function ◽  
Class Of Functions

We establish a new form of the3Gtheorem for polyharmonic Green function on the unit ball ofℝn(n≥2)corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functionsKm,ncontaining properly the classical Kato classKn. We exploit properties of functions belonging toKm,nto prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order2m.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

scholarly journals On the Green function of an orthotropic clamped plate in a half-plane

2021 ◽  
Author(s):  
Norbert Ortner ◽  
Peter Wagner
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Linear Combination ◽  
Half Plane ◽  
Orthotropic Plate ◽  
Dirichlet Boundary ◽  
Fundamental Solutions ◽  
Dirichlet Boundary Conditions ◽  
Mirror Point ◽  
The Green Function

AbstractFirst, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$ ( Q ( ∂ ′ ) - a 2 ∂ n 2 ) ( Q ( ∂ ′ ) - b 2 ∂ n 2 ) (where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$ ∂ ′ = ( ∂ 1 , ⋯ , ∂ n - 1 ) and $$a>0,b>0,a\ne b)$$ a > 0 , b > 0 , a ≠ b ) with respect to Dirichlet boundary conditions at $$x_n=0.$$ x n = 0 . The Green function $$G_\xi $$ G ξ is represented by a linear combination of fundamental solutions $$E^c$$ E c of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$ Q ( ∂ ′ ) ( Q ( ∂ ′ ) - c 2 ∂ n 2 ) , $$c\in \{a,b\},$$ c ∈ { a , b } , that are shifted to the source point $$\xi ,$$ ξ , to the mirror point $$-\xi ,$$ - ξ , and to the two additional points $$-\frac{a}{b}\xi $$ - a b ξ and $$-\frac{b}{a}\xi ,$$ - b a ξ , respectively.

scholarly journals Existence Results for Polyharmonic Boundary Value Problems in the Unit Ball

2007 ◽  
Vol 2007 ◽  
pp. 1-16
Author(s):  
Sonia Ben Othman ◽  
Habib Mâagli ◽  
Malek Zribi
Keyword(s):  
Unit Ball ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Elliptic Boundary Value ◽  
Nonlinear Elliptic ◽  
Nonlinear Elliptic Boundary

Here we study the polyharmonic nonlinear elliptic boundary value problem on the unit ballBinℝn(n≥2)(−△)mu+g(⋅,u)=0, inB(in the sense of distributions)limx→ξ∈∂B(u(x)/(1−|x|2)m−1)=0(ξ). Under appropriate conditions related to a Kato class on the nonlinearityg(x,t), we give some existence results. Our approach is based on estimates for the polyharmonic Green function onBwith zero Dirichlet boundary conditions, including a 3G-theorem, which leeds to some useful properties on functions belonging to the Kato class.

scholarly journals Ultrafunctions and Generalized Solutions

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Vieri Benci
Keyword(s):  
Dirichlet Boundary ◽  
Model Problem ◽  
Generalized Solution ◽  
Dirichlet Boundary Conditions ◽  
Generalized Solutions ◽  
New Class ◽  
Open Star ◽  
Space Of Distributions ◽  
Non Standard Analysis ◽  
Class Of Functions

AbstractThe theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think ofwith Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics (NAM) and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals ON A MINIMIZATION PROBLEM FOR A FUNCTIONAL GENERATED BY THE STURM – LIOUVILLE PROBLEM WITH INTEGRAL CONDITION ON THE POTENTIAL

2017 ◽  
Vol 21 (6) ◽  
pp. 57-61
Author(s):  
S.S. Ezhak
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Nonlinear Boundary ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Functional ◽  
Sturm Liouville ◽  
Class Of Functions

In this article we consider the minimization problem of the functional generated by a Sturm — Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions y and Q(x) isreduced to estimation of a nonlinear functional non depending on the potential Q(x). This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates are obtained for different values of parameter.

scholarly journals Existence of positive solutions for nonlinear boundary value problems in bounded domains ofℝn

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Faten Toumi
Keyword(s):  
Nonlinear Boundary ◽  
Nonnegative Function ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Boundary Value Problems ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Nonnegative Continuous Function ◽  
Bounded Below ◽  
Class Of Functions

LetDbe a bounded domain inℝn(n≥2). We consider the following nonlinear elliptic problem:Δu=f(⋅,u)inD(in the sense of distributions),u|∂D=ϕ, whereϕis a nonnegative continuous function on∂Dandfis a nonnegative function satisfying some appropriate conditions related to some Kato class of functionsK(D). Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous onD¯. Next, we will be interested in the Dirichlet problemΔu=−ρ(⋅,u)inD(in the sense of distributions),u|∂D=0, whereρis a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.

Non-existence of solutions for some nonlinear elliptic equations involving measures

2000 ◽  
Vol 130 (1) ◽  
pp. 167-187 ◽  
Author(s):  
L. Orsina ◽  
A. Prignet
Keyword(s):  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Model Problem ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Regular Functions ◽  
Nonlinear Elliptic ◽  
Bounded Open Subset ◽  
Image Position

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

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