An elliptic boundary-value problem with a discontinuous nonlinearity, II

1987 ◽  
Vol 105 (1) ◽  
pp. 23-36 ◽  
Author(s):  
G. Keady ◽  
P. E. Kloeden
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Step Function ◽  
Maximum Principles ◽  
Convex Domains ◽  
Heaviside Step Function ◽  
Elliptic Boundary Value ◽  
Image Position ◽  
Special Case

SynopsisLet Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have coresIn the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.

An elliptic boundary-value problem with a discontinuous nonlinearity

1981 ◽  
Vol 91 (1-2) ◽  
pp. 161-174 ◽  
Author(s):  
G. Keady
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Step Function ◽  
Discontinuous Nonlinearity ◽  
Elliptic Boundary Value Problem ◽  
Heaviside Step Function ◽  
Elliptic Boundary Value ◽  
Image Position ◽  
Special Case

SynopsisWe study the boundary-value problem, for(λ/k,ψ),Here ∆ denotes the Laplacian,His the Heaviside step function and one of A or k is a given positive constant. We defineand usually omit the subscript. Throughout we are interested in solutions with ψ>0 inΩ and hence with λ/=0.In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψtendsto zero when the area of Aψ, tends to zero.For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.

UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

2015 ◽  
Vol 58 (2) ◽  
pp. 461-469 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

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