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.

scholarly journals Eigenvalue characterization for (n · p) boundary-value problems

1998 ◽  
Vol 39 (3) ◽  
pp. 386-407 ◽  
Author(s):  
Patricia J. Y. Wong ◽  
Ravi P. Agarwal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Special Case

AbstractWe consider the (n, p) boundary value problemwhere λ > 0 and 0 ≤ p ≤ n - l is fixed. We characterize the values of λ such that the boundary value problem has a positive solution. For the special case λ = l, we also offer sufficient conditions for the existence of positive solutions of the boundary value problem.

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.

scholarly journals Eigenvalues of boundary value problems for higher order differential equations

1996 ◽  
Vol 2 (5) ◽  
pp. 401-434 ◽  
Author(s):  
Patricia J. Y. Wong ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Of Positive Solutions ◽  
Alpha Beta ◽  
Special Case ◽  
Higher Order Differential Equations

We shall consider the boundary value problemy(n)+λQ(t,y,y1,⋅⋅⋅,y(n−2))=λP(t,y,y1,⋅⋅⋅,y(n−1)),n≥2,t∈(0,1),y(i)(0)=0,0≤i≤n−3,αy(n−2)(0)−βy(n−1)(0)=0,γy(n−2)(1)+δy(n−1)=0,whereλ>0,α,β,γandδare constants satisfyingαγ+αδ+βγ>0,β,δ≥0,β+α>0andδ+γ>0to characterize the values ofλso that it has a positive solution. For the special caseλ=1, sufficient conditions are also established for the existence of positive solutions.

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.

scholarly journals Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

scholarly journals POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

2012 ◽  
Vol 86 (2) ◽  
pp. 244-253 ◽  
Author(s):  
YANG-WEN ZHANG ◽  
HONG-XU LI
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Existence And Uniqueness Theorem ◽  
Neumann Boundary Value Problem ◽  
Image Position ◽  
Continuous Dependence Of Solutions

AbstractIn this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

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