scholarly journals Numerical Enclosure-Methods for the Weak Solution of Elliptic Problems with Unilateral Constraints

1994 ◽  
Vol 186 (2) ◽  
pp. 306-337 ◽  
Author(s):  
M.I. Durucu ◽  
E. Kaucher
Keyword(s):  
Weak Solution ◽  
Elliptic Problems ◽  
Unilateral Constraints ◽  
Enclosure Methods

scholarly journals Existence results for nonlinear elliptic problems on fractal domains

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Massimiliano Ferrara ◽  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Critical Point ◽  
Elliptic Problems ◽  
Open Interval ◽  
Existence Results ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Fractal Domains ◽  
Critical Point Result

AbstractSome existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well-determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

scholarly journals Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Baishun Lai
Keyword(s):  
Weak Solution ◽  
Unique Solution ◽  
Biharmonic Equation ◽  
Final Section ◽  
Elliptic Problems ◽  
Weak Sense ◽  
Extremal Case ◽  
Singular Nonlinearity

We study the following semilinear biharmonic equation  , which satisfies0<u̲λ<1. In the extremal caseλ=λ*, we prove the existence of a weak solution which is the unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. We list those of particular interest in the final section.

scholarly journals Positive Solution to Singular Elliptic Problems with Subcritical nonlinearities

2019 ◽  
Vol 6 (1) ◽  
pp. 99-107
Author(s):  
Dharmendra Kumar
Keyword(s):  
Weak Solution ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Singular Elliptic Equations

AbstractIn this paper, we study the existence of a non-trivial weak solution to the following singular elliptic equations with subcritical nonlinearities:\left\{ {\matrix{ { - div\left( {{{\left| x \right|}^{ - 2\beta }}\nabla u} \right) - \mu {{f(x)u} \over {{{\left| x \right|}^{2(\beta + 1)}}}} = {{\lambda g(x)} \over {{u^\theta }}} + h(x){u^p}\,\,\,\,in\,\,\,\Omega ,} \hfill \cr {u > 0\,\,\,in\,\,\Omega ,} \hfill \cr {u = 0\,\,on\,\,\partial \Omega ,} \hfill \cr } } \right.where Ω ⊂ℝN is an open bounded domain with C1 boundary, θ, λ > 0, 0 < \beta < {{N - 2} \over 2} 0< p< 1, 0 < \mu < {\left( {{{N - 2(\beta + 1)} \over 2}} \right)^2}, N ≥ 3, 0 ∈ Ω and 0 ≤ f, g, h ∈ L∞ (Ω). We show that there exists a solution u \in H_0^1\left( {\Omega ,{{\left| x \right|}^{ - 2\beta }}} \right) \cap {L^\infty }(\Omega ) to this problem.

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 Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
B. Borsos ◽  
János Karátson
Keyword(s):  
Finite Element ◽  
Newton Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Elliptic Problems ◽  
Iterative Solvers ◽  
Finite Element Solution ◽  
Element Solution ◽  
Nonlinear Beam ◽  
Quasi Newton

Abstract The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.

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