Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions

2018 ◽  
Vol 45 (2) ◽  
pp. 1105-1128 ◽  
Author(s):  
Xiangyun Meng ◽  
Martin Stynes
Keyword(s):  
Convergence Analysis ◽  
Fourth Order ◽  
Two Dimensions ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Order Problem ◽  
Fourth Order Problem

scholarly journals A solution decomposition for a singularly perturbed fourth-order problem

Analysis ◽  
2017 ◽  
Vol 37 (2) ◽  
Author(s):  
Katharina Höhne ◽  
Sebastian Franz ◽  
Marcus Waurick
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Fourth Order ◽  
Singularly Perturbed ◽  
Third Order ◽  
Order Problem ◽  
Unit Square ◽  
Fourth Order Problem ◽  
Formal Power

AbstractWe consider a singularly perturbed fourth-order problem with third-order terms on the unit square. With a formal power series approach, we decompose the solution into solutions of reduced (third-order) problems and various layer parts. The existence of unique solutions for the problem itself and for the reduced third-order problems is also addressed. To our knowledge, this paper is a first attempt for a solution decomposition of such problems.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Some nonexistence theorems for semilinear fourth-order equations

2018 ◽  
Vol 149 (03) ◽  
pp. 761-779 ◽  
Author(s):  
M. Á. Burgos-Pérez ◽  
J. García-Melián ◽  
A. Quaas
Keyword(s):  
Fourth Order ◽  
Exterior Domains ◽  
Order Problem ◽  
Previous Condition ◽  
Fourth Order Problem ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Fourth Order Equations

AbstractIn this paper, we analyse the semilinear fourth-order problem ( − Δ)2 u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu &gt; 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ &gt; 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu &gt; 0 in ℝN, and they do not exist when the previous condition fails.

scholarly journals Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

2021 ◽  
Vol 26 (2) ◽  
pp. 227-240
Author(s):  
Lin Li ◽  
Donal O’Regan
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Fourth Order ◽  
Local Minimum ◽  
Elliptic Problems ◽  
Order Problem ◽  
Critical Nonlinearities ◽  
Fourth Order Problem ◽  
Minimum Theorem ◽  
Sublinear Perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

Combined Effects of Concave-Convex Nonlinearities in a Fourth-Order Problem with Variable Exponent

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Sami Baraket ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Fourth Order ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Direct Method ◽  
Energy Functional ◽  
Order Differential Operator ◽  
Order Problem ◽  
Fourth Order Problem ◽  
The Right ◽  
Fourth Order Differential Operator

AbstractWe study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes the existence of a nontrivial weak solution in the case of a small perturbation of the right-hand side. The proof combines variational methods, including the Ekeland variational principle and the mountain pass theorem of Ambrosetti and Rabinowitz. Next we consider a very related eigenvalue problem and we prove the existence of nontrivial weak solutions for large values of the parameter. The direct method of the calculus of variations, estimates of the levels of the associated energy functional and basic properties of the Lebesgue and Sobolev spaces with variable exponent have an important role in our arguments.

scholarly journals Remarks on solutions of a fourth-order problem

2006 ◽  
Vol 19 (7) ◽  
pp. 661-666
Author(s):  
Anne Beaulieu ◽  
Rejeb Hadiji
Keyword(s):  
Fourth Order ◽  
Order Problem ◽  
Fourth Order Problem
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