Study of the Singular Yamabe Problem in Some Bounded Domain of ℝn

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Yomna Rébaï
Keyword(s):  
Weak Solution ◽  
Bounded Domain ◽  
Boundary Data ◽  
Yamabe Problem ◽  
Positive Weak Solution ◽  
Regular Subset

AbstractIn this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝto have a positive weak solution in Ω with 0 boundary data, which is singular at each x

scholarly journals The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Giovany M. Figueiredo ◽  
A. Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Elliptic Equation ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Logistic Equation ◽  
Positive Weak Solution

AbstractIn this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

scholarly journals Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

2017 ◽  
Vol 6 (4) ◽  
pp. 447-459 ◽  
Author(s):  
Maya Chhetri ◽  
Pavel Drábek ◽  
Ratnasingham Shivaji
Keyword(s):  
Weak Solution ◽  
Weight Function ◽  
Bounded Domain ◽  
Exterior Domain ◽  
Additional Assumption ◽  
Exterior Domains ◽  
Singular Problems ◽  
Positive Weak Solution ◽  
Simply Connected ◽  
Decay Assumption

AbstractWe consider the problem\left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}} ({N>2}) is a simply connected bounded domain containing the origin with {C^{2}} boundary {\partial\Omega}, {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, {1<p<N} and {0\leq\delta<1}. In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution {u\in C^{1}(\overline{\Omega^{e}})} with {u=0} pointwise on {\partial\Omega}. Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in {\mathbb{R}^{N}}, for certain classes of {K(x)=K(|x|)}, we observe that our solution must also be radial.

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.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

On regularity criteria of a weak solution to the three-dimensional magnetohydrodynamics equations in Lorentz space

2018 ◽  
Vol 148 (3) ◽  
pp. 593-604
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Weak Solution ◽  
Bounded Domain ◽  
Lorentz Space ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
Magnetohydrodynamics Equations

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

NON-EXISTENCE, MONOTONICITY FOR POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC SYSTEM IN $\mathbb{R}_+^N$

2010 ◽  
Vol 12 (03) ◽  
pp. 351-372 ◽  
Author(s):  
YUXIA GUO
Keyword(s):  
Weak Solution ◽  
Elliptic System ◽  
Half Space ◽  
Positive Solutions ◽  
Integral Inequality ◽  
Existence Results ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Moving Plane ◽  
Positive Weak Solution

In this paper, by using the Alexandrov–Serrin method of moving plane combined with integral inequality, we prove some non-existence results for positive weak solution of semilinear elliptic system in the half-space [Formula: see text].

scholarly journals Singular Limits for 4-Dimensional General Stationary Q-Kuramoto-Sivashinsky Equation (Q-Kse) with Exponential Nonlinearity

2016 ◽  
Vol 24 (3) ◽  
pp. 295-337
Author(s):  
Taieb Ouni ◽  
Sami Baraket ◽  
Moufida Khtaifi
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Real Number ◽  
Decomposition Method ◽  
Smooth Boundary ◽  
Domain Decomposition Method ◽  
Exponential Nonlinearity ◽  
Singular Limits ◽  
Positive Weak Solution ◽  
Sivashinsky Equation

AbstractLet Ω be a bounded domain inwith smooth boundary, and let 𝓧1; 𝓧2; · · ·, 𝓧m be points in Ω. We are concerned with the singular stationary non-homogenous q-Kuramoto-Sivashinsky eaquation (q-KSE:where we use some nonlinear domain decomposition method to give a suficient condition to have a positive weak solution u in Ω under the physical Dirichlet-like boundary conditions, which is singular at each 𝓧ias the parameters λ, ϒ and ρ tend to 0 and where q ∈ [1, 4] is a real number.

scholarly journals On the definition of solution to the total variation flow

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
Juha Kinnunen ◽  
Christoph Scheven
Keyword(s):  
Weak Solution ◽  
Variational Inequality ◽  
Total Variation ◽  
Boundary Data ◽  
Duality Result ◽  
Total Variation Flow ◽  
Definition Of

AbstractWe show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals.

scholarly journals Which solutions of the third problem for the Poisson equation are bounded?

2004 ◽  
Vol 2004 (6) ◽  
pp. 501-510
Author(s):  
Dagmar Medková
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Bounded Domain ◽  
Linear Functional ◽  
Nonnegative Function ◽  
Newtonian Potential ◽  
Bounded Linear ◽  
The Third ◽  
The Poisson Equation ◽  
Bounded Linear Functional

This paper deals with the problemΔu=gonGand∂u/∂n+uf=Lon∂G. Here,G⊂ℝm,m>2, is a bounded domain with Lyapunov boundary,fis a bounded nonnegative function on the boundary ofG,Lis a bounded linear functional onW1,2(G)representable by a real measureμon the boundary ofG, andg∈L2(G)∩Lp(G),p>m/2. It is shown that a weak solution of this problem is bounded inGif and only if the Newtonian potential corresponding to the boundary conditionμis bounded inG.

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